IN THIS ISSUE— Vol. 12, No. 3 March, 1962 - Mathematical ...

PacificJournal ofMathematics

IN THIS ISSUE—Alfred Aeppli, Some exact sequences in cohomology theory for Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791Paul Richard Beesack, On the Green’s function of an N-point boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801James Robert Boen, On p-automorphic p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813James Robert Boen, Oscar S. Rothaus and John Griggs Thompson, Further results on p-automorphic p-groups . . . . . . . . . . . . . . . . 817James Henry Bramble and Lawrence Edward Payne, Bounds in the Neumann problem for second order uniformly elliptic

operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823Chen Chung Chang and H. Jerome (Howard) Keisler, Applications of ultraproducts of pairs of cardinals to the theory of

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835Stephen Urban Chase, On direct sums and products of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847Paul Civin, Annihilators in the second conjugate algebra of a group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855J. H. Curtiss, Polynomial interpolation in points equidistributed on the unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863Marion K. Fort, Jr., Homogeneity of infinite products of manifolds with boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879James G. Glimm, Families of induced representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885Daniel E. Gorenstein, Reuben Sandler and William H. Mills, On almost-commuting permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913Vincent C. Harris and M. V. Subba Rao, Congruence properties of σr (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925Harry Hochstadt, Fourier series with linearly dependent coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929Kenneth Myron Hoffman and John Wermer, A characterization of C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941Robert Weldon Hunt, The behavior of solutions of ordinary, self-adjoint differential equations of arbitrary even order . . . . . . . . . . . 945Edward Takashi Kobayashi, A remark on the Nijenhuis tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963David London, On the zeros of the solutions of w′′(z)+ p(z)w(z)= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979Gerald R. Mac Lane and Frank Beall Ryan, On the radial limits of Blaschke products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993T. M. MacRobert, Evaluation of an E-function when three of its upper parameters differ by integral values . . . . . . . . . . . . . . . . . . . . . 999Robert W. McKelvey, The spectra of minimal self-adjoint extensions of a symmetric operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003Adegoke Olubummo, Operators of finite rank in a reflexive Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023David Alexander Pope, On the approximation of function spaces in the calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029Bernard W. Roos and Ward C. Sangren, Three spectral theorems for a pair of singular first-order differential equations . . . . . . . . . . 1047Arthur Argyle Sagle, Simple Malcev algebras over fields of characteristic zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057Leo Sario, Meromorphic functions and conformal metrics on Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079Richard Gordon Swan, Factorization of polynomials over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099S. C. Tang, Some theorems on the ratio of empirical distribution to the theoretical distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107Robert Charles Thompson, Normal matrices and the normal basis in abelian number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115Howard Gregory Tucker, Absolute continuity of infinitely divisible distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125Elliot Carl Weinberg, Completely distributed lattice-ordered groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131James Howard Wells, A note on the primes in a Banach algebra of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139Horace C. Wiser, Decomposition and homogeneity of continua on a 2-manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145

Vol. 12, No. 3 March, 1962

PACIFIC JOURNAL OF MATHEMATICS

EDITORSRALPH S. PHILLIPS A. L. WHITEMAN

Stanford University University of Southern CaliforniaStanford, California Los Angeles 7, California

M. G. ARSOVE LOWELL J. PAIGE

University of Washington University of CaliforniaSeattle 5, Washington Los Angeles 24, California

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SOME EXACT SEQUENCES IN COHOMOLOGY

THEORY FOR KAHLER MANIFOLDS

ALFRED AEPPLI

1Φ Introduction* In this note some results announced in [1]concerning exact sequences for Kahler manifolds are proved. The mainresult is that there exists an exact sequence relating the usual bigradedgroups of harmonic forms on a compact Kahler manifold K and animbedded submanifold L with certain mixed relative cohomology groupsof (K, L). These mixed cohomology groups have been introduced byHodge [6]. Using the results of Hodge the exactness of the givensequence is derived in a straightforward manner. H. Guggenheimerconsidered in [5] such a sequence too. Finally the exact sequence isapplied to deduce some results concerning the imbedding of a complexmanifold L in a Kahler manifold K, in particular the following statementis proved: if the imbedding L c K is homology faithful, then br>s{K) =br>8(L) implies br-kt8-k(K) = 6r_Λiβ_Λ(L) for k = 0,1, 2, (br>s = rank ofthe module of harmonic (r, s)-forms). The paper is organized the followingway: § 2 gives the necessary notations and a few known results; § 3contains the main theorem (Theorem 1) the proof of which is given in§ 4; some applications follow in §§ 5 and 6, in particular Theorem 2which implies the foregoing statement on homology faithful imbeddings.

2 Notations and known results*(a) We use the following notations:Fr'8: group of the complex valued C°°(r + s)-forms of type (r, s) on

a complex manifold M = M2n.Fk = Σ Fr's-

r+s=k

d = d' + d": exterior differentiation operator on M; d: Fk —• Fk+1,d':Fr s->Fr+1 s, d":Fr s->Fr's+1 (cf [2], [6]).*: Fr s —> Fn~s n~r: Hodge—de Rham duality operator which as-

sociates to a its (metrically) dual form *#, with the help of a Hermitianmetric (of class C°°) on M (cf. [6] and [3]; ** = (- l )* ( a -* ) = ( - l ) * : Fk-*Fk).

8 = -*<Z* = δ' + δ" = - ( * d " * + *d'*) (<?' = -*<Γ* resp. δ" = - * d ' *is an operator of type (—1, 0) resp. (0, —1)).

V = d'd", V* = δ"δ' = (~1)Λ + 1*F*.J = ώδ + δd: Laplace—Beltrami operator.A' = d'5' + δ'df, Δ" = d"δ" + δ"d".We define Zy = {a\ae Fr % d'a = 0}, similarly Zy, ZF

r'% Zr

d's = Zj/,V,,

Received October 10, 1961. This work was supported by a grant of the National ScienceFoundation (N. S. F. Project G-15985, Cornell 6126).

791

792 ALFRED AEPPLI

moreover Ry = {a\ae Fr>s, a = d'β for some β} = d'F*-1'8, similarlyRy = d"Fr'-\ R1

F

S = FF1-1--1, Ryd,, = d'F*-1'" 0 d"Fr s-\ Ry ={a\ae Fr s, a = dβ for some β}, and dually Zζ f etc1. Let Hy = Zy bethe group of harmonic (r, s)-forms, Hy = Zr

Δ;s and Hr

Δ;f = Zr

Δ;f.Considering the components of different type in dda, the relation

dd = 0 implies

(1) d'd' = 0 , d"d" - 0 , d'd" + d"d' = 0 ,

and therefore the following relations hold:

(2)

T>r,s ,— f/r.s J?r,s ,— 'Vr.s

r,s f— ~Dr,s r\ T?r,s r~\ Έ}r,s ,— Π?'r,s ΠΓr^ Φr ,s r\

/ C Kd) 11 Kdn 11 Kd C Zjd' — Zid'.d" — ^d> IIγ s ,.— ~Dr,s ΐ?f s fΊ\ T-?*"'8 f— Γ7T s *P\ Γ7r s i— ΠΓv sd {— •**&' ,d" — JX'dl \D -Hd" {— Ad' \D ^d" ^ ^V

(2) enables us to form the cohomology groups Hy = Zj '/Ry,Hr

d;f = ZtflRy, (Dolbeault groups), and Hr

dfv = HyΛ,Ίv = Zy\Ry, H;fd,,d,, =Zr

ψ'*\Ry>d,, (mixed cohomology groups). Dually to (1), (2), there arerelations δ'δ' = 0 etc., Ry c Z[;s etc., and cohomology groups iϊδ

r;s etc.(b) If M is a Kahler manifold (cf. [2], [6], [4]) one has

(3) Δ = 2Δf = 2Δ" ,

(4) d'δ" + δ"d' = 0, d"δ' + δ'd" - 0 ,

such that HA — HA, = HA,, and (using the obvious notation) Zd^>> = Z8,,a,,Zd»v — Zva>>. Moreover, (4) makes it possible to consider the furthermixed cohomology groups Hd;*vΊd,h,., Hd;i»ld,tV,, Hl:JthΊd,,v, Hd/?δΊd,,>8, (using

again the obvious notation).For a compact Kahler manifold M the following isomorphisms (listed

for completeness sake) are well known: Hy — HΔ

r (Eckmann-Gug-genheimer), Hy = H^~s,n-r ( r efln e ( j Poincare duality; holds for anycompact complex manifold with a Hermitian metric). The de Rham—Hodge theorem

(o; -a* = J±d ~ iij = 2 J J*Δ'

will be used later on. Here Hp is the usual pth cohomology group ofM over the field C of the complex numbers and Hi the ordinary pthde Rham cohomology group of M over C. The most important resultsfrequently used in the sequel are the isomorphisms proved by Hodgein [6]:

(6) Hy = Hy = Hl f ,

V ' / ±Λ-Λ = •Lίdl7 t

1 {(etc.)) indicates—here and sometimes in the sequel—the possibility of analogous formulasor expressions using dual and/or complex conjugate operators.

SOME EXACT SEQUENCES IN COHOMOLOGY THEORY 793

(8) Hy = Hft v, .

Dually Hy = Hζ;? etc. [6] contains also Hy ^ Hifaiw = Hl.lΊd.Λ,. ~Hd>'?,8>id"v = fld''V/(ί".8'. The proofs for these isomorphisms and for (6),(7), (8) are strongly related; one applies the decomposition theorem forthe forms on a compact Kahler manifold, deduced with the help of (3)and (4). The considered isomorphisms are induced in a natural way byinclusion {Zy c Zy and so on).2

3* The main theorem*(a) Let K — K2n and L = L2m be compact Kahler manifolds, m < n,

and let L be imbedded in K, regularly (and complex analytically) suchthat there exist coordinates z19 z2, * ,zn in the neighborhood U(x) c Kof any point x e L with L Π U(x) = {p | p e U(x), zλ(p) — = zn-m(p) — 0}We consider the relative groups Frs(K, L) = {a\a e Frs(K), a = 0 alongL], ZJ;S(K, L) - {a\aeFr %K, L), d'a = 0} = Zy(K) Π F r s(iΓ, L) etc.,jβj;s(iί, L) = d'Fr-ls(K, L) etc. ((a = 0 along L)) means: α: vanishes

tangentially along L. The inclusion L > K induces homomorphisms

Frs(K)-^Frs(L), Frs(K,L)-^>Frs(K), and i ,i* commute withcί', cZ", F. Clearly the relations (2) hold still in the relative case, so therelative cohomology groups Hy(K, L), Hy(K, L), HfiftK, L), H;ιl.td,,{K, L)are well defined: Hy(K, L) - Zy(K, L)IRr

d,s(K, L) etc. Using (6), (7), (8),

the operators i*, β, d', d", F induce the cohomology homomorphisms ΐ*,j*, dr, d", V in the following sequences (cf. [5]):

— Hy(K,L) JL>H

(10) ίl> iϊ;.-i(L) - ^ J?ί;f(if, L) -^> Hy(K) -^> ί β (L)

(11) %1+ HΓ'ΛL) - ^ HlifrK, L) -£+ Hy{K)

ΐ * , ί * are given in the natural way; d\d'\V are explained in (b).In §4 we prove the

THEOREM 1. The sequences (9), (10), (11) are exact.

We will treat only (11); the considerations can easily be carried overfor to get the analogous results for (9) and (10)3. In the sequel, all theoccurring sequences of groups and homomorphisms are exact.

2 It follows from (6) that dim HJ S is independent of the Kahler metric on M.~Hj ° = Hp? is an equality JEΓJ" ° = Hfc? = Zfi? and says: every holomorphic r-form is aharmonic (r, 0)-form and vice versa.

3 The constructions in (9) and (10) and the proof for Theorem 1 in these cases arecompletely parallel to the corresponding procedures in the case of the de Rham cohomology(using the operator d). Hence we are mostly interested in (11).

794 ALFRED AEPPLI

(b) To describe the homomorphism 7 in (11) we follow the usualprocedure as it can be found e.g. in [7], p. 192. Every element a e Frs{L)is the trace of an element a e Frs{K), and one has

0 > FT %K ,L) -JL Frs(K) -^-» Fr (L) > 0 .

Applying 7, we get the commutative diagram

0 * Frs(K,L) -?-+ F'- (K) -^-> Frs(L) >0

(12)

0 > Fr+1 3+1(K, L) -£-+ Fr+1 3+1(K) - ^ Fr+X '+1{L) » 0

Let a e Z ; S(L), and let α = i*&, & e Fr '(K). Then iΨa = 7i*ά =

7a = 0, i.e. 7a 6 i?"+1 + 1 ( ^ , L). d'7 = d"7 = 0 implies Fα 6 Z,Γ+1 + 1(Z, L).

Let jδ be a second extension of a: a = i*β, β 6 Fr '{K). Since F(α - /3) =

7a — 7β, a — β — 0 along L, F« and 7β determine the same element in

Hίft-'+\K, L) such that we get a homomorphism Zr,\L)-^ H^hs+1(K,L).

Moreover Ry,ΛL) - ^ 0 since every form a e Rr

d;s

d,<(L) is trace of a form

ά e RyΛ-(K) (and because Γd' = Fd" = 0). Therefore, (12) gives rise to

a homomorphism H;j ά.Λ.,(L) - ^ mϊ}Λ$(K, L) = H^*+1(K, L) and to

the sequence

(13) ...-£-* Htf.tf{L) -?-> Hl&K, L)

Hϊ {K)

with i* = i I where H;s(K)-^ Hr

r,s

d,,d,,(K) is the canonical isomorphism

(8) and # ; ^ , ^ ( i θ — ^ H;;lΛ..(JL) is the restriction homomorphism be-

longing to the inclusion ί, and with j* — Jj where Hdβ(K, L) •

Hϊ ϊ(K) is the inclusion homomorphism belonging to i and HZ/&K) •

Hy(K) is the inverse of the canonical isomorphism (7). Replacement

of H;ιίtd,.(L) by JBΓ (L) in (13) in virtue of (8) leads to the sequence

(11), and in order to prove exactness of (11) it is enough to prove

exactness of (13).

4 Proof of the main theorem*

(a) H;ri'*p{L) -£-> HSifrK, L) - ^ Ht


Fάx = a, i.e. im(F) D ker(j).

(b) HlfXK, L) -L Hjrf(K) — HϊfF(L) .

Proof, ί] = 0 immediate. Let α e Hd,s

F(K) with ?α = 0, representedby α. Then ΐ*α = Fa' for an α' e Fr~ls-\L). Let α' be an extensionof a! (i*af = α') such that /9 = a — Fά'eZd'

s(K, L). /3 represents/3 e H£js

F(K, L), and J ^ = α. This shows im(J) 3 ker(ί).

(c) H^AK) - ^ H;,ί.d.t(L) -Ϊ-+ H^'s+1(Kf L) .

Proof. Fi = 0 immediate. Let a e Hfiίtd»(L) with Fά = 0, repre-sented by #. Let ί#α = a, a e Fr S(K). β = Fa represents then thezero element in Hr

d^'s+\K, L); therefore β = Fa, for an a,eFrs(K, L).Hence 7 = a — aλ e ZF

S(K), y represents an element 7 e HFis

d>d/{K)y and%Ί — a. Therefore im(i) ZD ker(F).

(d) We have the commutative diagram

\hL

where hκ, hL are naturally induced isomorphisms (by [6], cf. (7) and (8)).(a), (b), (c), (d) and (7), (8) imply exactness of (13) and (11).

5. Some remarks and first applications*(a) REMARK 1. If if is a Kahler manifold, and if the complex

manifold L is complex analytically immersed in K, then L is a Kahlermanifold too since the Kahler metric in K induces one in L. So Theorem1 holds if if is a compact Kahler manifold and L a regularly imbeddedcomplex manifold in K.

REMARK 2. The homomorphism H;j8

d,td,,(L) -?-* H^-'^K, L) can bedefined also in the following situation: K and L complex manifolds ofthe same dimension, L open subset of K. One makes use of the localcharacter of the operator V (i.e. TFa c Ta for Ta = carrier of a).Inclusion and restriction homomorphisms are defined the usual way, and(a), (b), (c) hold in this case too.

REMARK 3. (9), (10) hold in the form

796 ALFRED AEPPLI

(10-) flU Hlf~\L) -*L> HV(K, L) -JU Hl;f(K) -^-> H

for complex manifolds K, L (Kahlerian or not) with dim L < dim K andL regularly imbedded in K, or with dim L = dim K and L = open subsetof K.

REMARK 4. An analogous dual construction in the situation L — opensubset of K is possible for §', δ" where one gets exact sequences dualto (9'), (10"). J7* gives rise to the dual statements of (a), (b), (c) (againfor L = open subset of K).

(b) We return to our original assumptions as stated in § 3 (a). Weput HI = Σr+.-pfl2>β etc., and (5), (6), (7), (8) imply

H>(K) s H!{K) s HI(K) = H$,(K) ~ Hl.(K) s fliWίΓ) ~ H>ld,td,.(K)

for a compact Kahler manifold K. Moreover, considering the exactsequences (9), (10) besides those which belong to the de Rham cohomologyand to the usual cohomology, we get ([1], p. 293)4

(14) H%K, L) s H!(Kf L) ~ Hl(K, L) s Hϊ,(K, L) .

It seems harder to express HSιΨ(K, L) (in which we are interestedbecause of (11)) by means of known quantities. A formula similar to(14) is certainly not correct for these groups. We have the commutativediagram

α5) i .* i ., i.. .-£-,tf;-i..-i(L) > HlfAK, L) > H r ^

which implies Hr

dί;{K, L) = dr Hrls~\L) 0 j * Hr

ά,f(Ky L), i.e.

H;f F(K, L) 0 i * Hϊ-\K, L) ^ Hfi-\K, L)®j* Hl'!{K, L) .

Moreover, (15) implies together with the diagram obtained from (15)by interchanging dr with d" and adjusting the dimension indices

= d/f Jϊ (L) ,

j*Hϊ;ϊ(K, L) s j*Hi; (K, L) s j*Htf(Kf L)

such that

4 As always, we are interested only in the additive structure of the different cohomologygroups, so the isomorphisms are understood to be additive. The investigation of themultiplicative structures—where such are possible—is interesting in connection with Poincareduality and has not yet been carried out in great details for the situation considered inthis paper.


m,1{K, L) ~ d"Hr-\L)@j*m;?{K, L) - d'Hϊ-\L)@ j*m?{K, L) ,

andm,s{K, L) 0 d'Hϊ-\L) = m?(K, L) 0 d"HΓls{L) .

(c) A very simple application of (10") is the following. We have

0 > H£?(K, L) -^U Hj,?(K) - ^ H;, ?(L) — ,

and suppose H£?(L) = 0, i.e. Hj°(L) = 0 in the case of a compact Kahlermanifold L (6r,0(L) = 0, where 6r>s = ά\m0H

r

Δ

s). Then Hτ

d,?(K, L) ~ Hr

d,?{K),this isomorphism being induced by inclusion, saying that every holomorphicr-form in K vanishes along L. This also follows immediately fromHl?{L) = 0. As a consequence, the complex protective space P ( m ) cannotbe regularly imbedded in a complex torus T{m+m (with a complex paral-lelizable structure), N 0. The same is true for any complex manifoldL(m) instead of P ( m ) with Hj;?(L) = 0 for some r, B r ^ m , Anotherexample: let L be again a complex manifold with H^(L) = 0 for somer, 1 r ^ m = dim^L), and consider I = L x T(2sn, N^r, as a fibrebundle with base L and fibre T. Then the only complex analytic crosssections in M are given by L x p, pe T{N).δ

6Φ Further applications to the imbedding L c K. In a compactKahler manifold K ~ K{n) multiplication with the fundamental class ωinduces according to [4]

(16) 0 > HΓ1S~\K) — HJ '(K) for r + s ^ n

(ω is the cohomology class belonging to the form induced by the Kahlermetric). Considering the commutative diagram (in which we use (16))

I *Tjr—1 s—1/ 17* T \ TJr—

!

0

where J* is also induced by multiplication with ω, the condition

(-\Π\ 0 TJr—1,8—1/ Tζ T\ J

>

yields

5 Added in proof. Cf. S. S. Chern, Geometrical Structures on Manifolds, Coll. LecturesSummer Meeting AMS, Michigan 1960, pp. 26-30, for further remarks on the possibilitiesof holomorphic maps /: V-^M.

798 ALFRED AEPPLI

(18) 0 > Htf'-^K, L) — HϊifrK, L) .

In view of (11) the condition (17) is equivalent to

(19) HΓ2 S~XK) - ^ HΓ*'-\L) > 0 ,

and we get the

LEMMA. / / K, L is a pair of compact Kdhler manifolds, L regularlyimbedded in K, then (19) implies (18) (for r + s ^n = dim^ K).

If in addition HΓls'\K) ^-~> HΓls'\L) >0 and 0 >H# J S(L), then (by (11)) Hί frK, L) = 0, and the Lemma implies Hfc1'-1 = 0

such that (again by (11)) 0 > Hrls'W) ~^—> HΓls~\L) > 0, i.e. weproved the

THEOREM 2. / / K, L is a pair of compact Kdhler manifolds, Lregularly imbedded in K, and if for some r,s, r + s^n~ dimσ Kf 0 >

Hy(K) -ί-> HJS(L), HΓ'rW) ^HT^ΛL) > 0, HΓ*'-\K) - ^

HΓ2SΛL) >0, then HΓl8~\K) ~ -> Hrls~\L) is an isomorphism(and therefore b^ltS^(K) = br_ltS_γ(L)).

Repeated application of Theorem 2 leads to

COROLLARY 1. K, L, r, s as in Theorem 2. Let 0 > Hr

Δ

s(K) •

HJ*(L) and HΓk*~k(K)-^> HΓks~k(L) >0 for fc = 1, 2, •••, thenthese epimorphisms are isomorphisms (and therefore br^k,s^k(K) —br_M(L)for & = 1,2, . . . ) .

COROLLARY 2. if, L, r, s as above. Let the imbedding L c K be

homology faithful (i.e. H%υ(K) - —> H%V(L) > 0 for all u, v). Then

0 > Hϊ {K) -^-> Hr(L) implies

0 > HΓks-k(K) — Hrks~k(L) > 0 /or fc = 0,1, 2, .

(or: br>s(K) - 6,S(L) implies br^,s^(K) - δr_,,8_fe(L)).

EXAMPLE. The Poincare polynomial of P{m) x P ( w ) is

/7(P(m) x P(m)) = 1 + 2ί2 + 3£4 + + (m + l)ί2m + + 2£4m~1 + ί4w .

Let Pln) (n ^ 2m) be the complex manifold which we get from P{n) bymeans of ^-modifications at m different points. Pίn] is a compact Kahler(even algebraic) manifold with

Π(Pln)) = l + (m + l)ίa + (m + l)ί4 + . . . + (m + I)ί2w~2 + t2n .


Corollary 2 implies: there is no homology faithful regular imbeddingp(m) χ pim) c P(»,β T h e r e a r e r e g u l a r imbeddings of P{u) x P{υ) in

p(uv+u+v)f therefore in P±n) tor n ^ uv + u + v. It follows: these imbed-

dings are not homology faithful. Similar results hold for P ( m ) x p^ciP{n),

P{n) obtained from P{n) by (/-modifications at q different points, 1 ^ q ^ m

REFERENCES

1. A. Aeppli, Modifikation von reellen und komplexen Mannigfaltigkeiten, Comm. Math.Helv, 3 1 (1956), 219-301.2. H. Cartan Seminaire 1951-52, ENS Paris. Expose I.3. G. de Rham, Varietes differentiates , Hermann, Paris 1955.4. B. Eckmann et H. Guggenheimer, Sur les varietes closes a metrique hermitienne sanstorsion, C. R. Acad. Sci. Paris, t. 229, (1949), 503-505.5. H. Guggenheimer, Topologia differenziale delle trasformazioni cremoniane e delle rieman-niane di funzioni di piύ variabili complesse. Convegno internazionale di geometriadifferenziale, Italia, (1953), 222-228, Roma 1954.6. W. V. D. Hodge, Differential forms on a Kdhler manifold, Proc. Cambridge Phil. Soc,47 (1951), 504-517.7. N. Steenrod, The topology of fibre bundles, Princeton University Press 1951.

ON THE GREENS FUNCTION OF AN TV-POINTBOUNDARY VALUE PROBLEM

PAUL R. BEESACK

1* Introduction* In a recent paper [3], D. V. V. Wend made useof the Green's functions g2(x, s), g3(x, s) for the boundary value problems

u" = 0; ΐφi) = u(a2) = 0, (αx < αa) ,

u'" = 0; u(ax) = u(a2) = u(a3) = 0, (αx < α2 < α3) .

In particular, he showed that if aλ ^ 0, then

I g2(x, s) I < α2 , | flr3(α, s) | < αl

for αx g x, s <L α2 or ax ^ a;, s ^ α3 respectively. He conjectured that ifgn(x, s) is the Green's function for the boundary value problem

(1.1) u{n) = 0; u{aΎ) = ... = u(an) = 0, (a, < α2 < < an) ,

then

I gn(x, s) I < al~\ aλ^x,s^an ,

(if aλ ^ 0) and states in a footnote that this conjecture has been verifiedfor n < 6. Assuming this conjecture valid he uses the inequality toobtain a lower bound for the mth positive zero of a solution of thedifferential equation

(1.2) y{n) +f(x)y = 0

where f(x) is continuous and complex-valued on 0 <£ # < °°. In this proof,all zeros of the solution are counted as though they were simple zeros.

In this paper, we consider a more general boundary value problemallowing for multiple zeros of y(x). Let gn{x, s) now denote the Green'sfunction of the differential system

α 3) \W ) tfifl) y"(a) yΛi) (μt) = 0 .

where ax < a% < < ar, 0 ^ kiy kx + k2 + + kr + r = n. In §2, weshall prove that

r

Π | τ π |

(1.4) \ ( ) \ ^' — ' " - (n-l)l(ar-ai) ~ V n I n\

for aλ< x, s < ar. In the case r = n, Wend's conjecture is thus verified,

Received September 28, 1961.

801

802 PAUL R. BEESACK

and improved. In §3, we apply this inequality to differential equationsof the form (1.2), and more generally to nonlinear differential equations

(1.5) y{n) +f(χ,y,yf, . . . f i/(-1}) = 0 ,

to obtain lower bounds for the mth zero of solutions. The inequality(1.4) also leads to an extension of an oscillation criterion of Liapounofϊfor the case n — 2.

2. The Green's function* If gn{x, s) denotes the Green's functionof the system (1.3), then gn satisfies—and in fact, is defined by—thethree conditions:

1° Qny 9n, , 9(n~2) are continuous functions of (x, s) on the squaredit^x, s ίg αr, while g{n~ι) is a continuous function of (x, s) in each ofthe two triangles ax ^ x ^ s ^ ar and αx <L s ^ x gΞ ar with

g(Γ1] (β + , s) - g{

n

n~1] (s-, s) = - 1 , a, < s < ar .

2°. #iw) (ίc,s) = 0 in each of the two triangles above.3°. For each s, with ax < s < ar, gn(x, s) satisfies the n boundary

conditions of the system (1.3).In the above statements (and throughout this paper), all derivatives

are taken with respect to x. For a thorough discussion of Green'sfunctions for much more general systems than (1.3), see Toyoda [2].The existence of gn depends on the fact that the system (1.3) is incom-patible. We need not verify this directly since the result will followfrom our method of proof which is by induction.

Although the conditions l°-3° define gn on the square a1^x9 s ^ αr,we want to extend the domain of definition of gn to the entire plane.We assert that this can be done in such a way that

( L ) gn, g'n, •• ,#ίΓ~2) are continuous for all (x, s), while g ^ iscontinuous in each of the half-planes x ^ s and s <£ x, with g{χ~1] (s + , s) —flrίΓ"1} (β—, s) = - 1 , - c o < s< ex..

( ΐ l n) g{n} (x, s) == 0 in each of the above half-planes.(IΠ») For each s, (— oo < s < co), gn(χ, s) satisfies the w boundary

conditions of the system (1.3).(IVW) gn(x, s) = 0 if s ?g min (alf x), or s ^ max (αr, x).We proceed to prove these assertions by induction. Suppose they

are valid for any system of the form (1.3). If aά is any zero of aboundary value problem of this form for the equation y{n+1) — 0, thecorresponding set of boundary conditions is either of the form (1.3) withkj replaced by kj + 1 (in case aό is not a simple zero for the new system),or is of the form

α,) = 0 ,

GREEN'S FUNCTION OF AN ΛΓ-POINT BOUNDARY VALUE PROBLEM 803

where now kλ + + fc,-i + kj+1 + + & r + r = w + l . Let gn+1 {x, s)be the Green's function for this new system. We assert t h a t

(2.2) gn+1(x, s) = — Ux - s)gn(x, s)n I

_ (α, - »)gJN+»(»,, s) {χ _ ή / x - gt y.+n(fey + 1)! j=i\ α, — α^/ J

in the first case noted above, while

(2.3) gn+1{x, s) = 1 {(s - «)(/.(», s) - (α, - s )^(α, , s

in the second case. Note that (2.3) is formally included in (2.2) by settingkj = - 1 in (2.2). In the sequel we work with (2.2) only; (2.3) will followby making use of this formal identity. We remark that gn is definedby the conditions l°-3° in 2(r — 1) ' 'pieces'', an explicit determinationof any "piece" requiring the solution of n nonhomogeneous linearequations. For this reason, the recursion relations (2.2), (2.3) may beof some interest in themselves.

For brevity, set

wη^'sUχ-a^ π)! *

For each s, P(x, s) is a polynomial of degree n in x. If k3- < n — 2 itfollows from our induction assumptions that P, as well as all its derivativeswith respect to x, is a continuous function of (x, s) in the entire plane.This also holds if k3? = n — 2 because of the factor (α3- — s), providedwe define P{x, a3) = 0. We also note that

P{m)(aif s) = 0, 0 ^ m ^ ki9 i Φ j ,

(2.4) P ( m ) ( α y , β) = 0, 0 ^ m ^ fcy ,

ah s) = (α y - β)flr<*'+1)(αy, β) .

(In the case k5 = — 1, the second of the identities (2.4) does not appear.)Differentiating (2.2) partially with respect to x, we obtain

flfi+ifc, s) = — {(a? - s)flfi(aj, s) + flfw(a?, s) - P'(α?, s)} ,

ΛVi(», s) = - {(x - β)ffϊ(», β) + 2g&x, s) - P"(^, β)} ,n

= — {(a? - βtoίΓ'ίs, 8) + m ^ - 1 } ( ^ , s) - P ( m )(x, 8)} ,n

1 ^ m ^ n + 1 .

804 PAUL R. BEESACK

By our induction assumptions, together with the preceding remarksconcerning P(x, s), it follows that gn+1, gή+i, •• ,flΰ+ϊ2> are continuous inthe entire plane. The same is true of g{n+i\ because of the factor(x — s). Finally, for x Φ s,

9 s) = -L {(x - s)g{:){xi s) + ng[:~ι)(x, s) - PM(x, s)} ,n

so that

1^n

f s) - g{

n

n-1](s-, s)} = -

and condition (/n+1) is thus satisfied. Condition (Πn+1) is also satisfiedsince g™ = 0 and P{n+1) Ξ O in each of the two half-planes x ^ s ands ^ x. For the boundary conditions we have

gimλ(aif s) = l {(α, - s)glT\aίf s) +n

a,, s) - P^(aif s)} = 0

for 1 g i g r and 0 g m g kiy using the first two of (2.4). Using thelast of (2.4), we also see that ff'fα^sjΞO, and (ΠIn+1) is satisfied.Finally, suppose s S min (alf x) so that gn{x, s) = 0, and hence alsognJ+1)(ajf s) = 0 since s ^ αx ^ αJβ Thus, by (2.2), the first of conditions(IVn+1) is satisfied, and similarly, so is the second.

For n — 2, we have explicitly

— oo < s

(2.5) , β) -

/γ QtΛ/ O ,

0,

(x — c

a.

(s-a

a.

0,s — x,

Ϊ — Λ]

i — 0]

- β )>

-x)

L

ΛJ ^ S,

S ^ X,

X ^ 8,

S ^ X,

X ^ S,

^ 8 ^ α 2 ,

α2<

from which one easily verifies that the conditions (I2)-(IV2) are satisfied,thus completing the induction for all n ^ 2.

Our goal now is to obtain an upper bound for \gn(x,s)\. It will,however, be easier to work with the related function Gn(x, s) defined by

(2.6)gn(x, s) = Gn(x, s) Π (x

ifor x Φ α« .

(α4, s) = (fci + 1)! Π (di - dm)km+1 Gn(aiy s) for x = at .

We note that for each fixed s Φ aif Gn(x, s) is continuous for all x. If

GREEN'S FUNCTION OF AN ΛΓ-POINT BOUNDARY VALUE PROBLEM 805

k{ < n — 2, Gn(x, at) is also continuous for all x, while if ki — n — 2,Gn(x, a,) has a finite jump at x = a{ and is otherwise continuous. (Thecase kj = —1 is again included in (2.6), the factor (x — α,)fc^+1 becomingunity in this case.) We now have

gn+1(x, s) = Gn+ί(x, s) (x - as) U(x- α<)*«+1, x Φ a, .ΐ = l

Using (2.2) and (2.6), we also have

= 1 {(a- - 8 ) Π (a? - a%

= I n (a? - α,m

n{x, s) - (αy - s) f[ (x - ax)^Gn{ah s)\

(», β) - fay - e)Gw(a i f β)} ,

so that

(2.7) (x - aj)Gn+1(x, «) = i {(x - s)Gn{x, s) - (a, - s)Gn{ah «)} .

We note that for all n ^ 2, (IVTC) gives

(2.8) Gn(x, s) Ξ 0 for s g min (α1; x) and s ^ max (ar, x) .

We now prove by induction that

1 1

(2.9)

(n — 1)1 ar — x '

1 1(n - 1)! ar-a,'

1 1

=oo < s < alf x ^

- 1)! x - αx ' α r < s < oo, s ^ a;.

For w = 2, (2.5) gives

\G%(X,8)\ =

( « 1

( « 2

( a ? -

(X

(s — x- x) (α2

- x) (α2

(β-α,

- aλ) (α2

-αθ(*

- x ) '

- O i ) '

v t

-a,)'

X ^ 8,

α2 ,

α 2 < s < o o , s ^ Ξ s c ,

from which (2.9) is immediately verified for n = 2.We will ^ϊrsέ prove (2.9) under the assumption that gn+1 has at

806 PAUL R. BEESACK

least three distinct zeros. Taking j = 1 in (2.2) and (2.7), we have for— oo < s < a19 x ^ s,

G m + 1 (» , β) =

by (2.8); hence

) = i- ULZJ- Gn(x, s) - -^—s- Gn(aly s)\n I x — α x # — α 2 J

w aλ — a? w! α r — x

Similarly, taking j = r in (2.2) and (2.7) we obtain for α r < s < oo, s <£ #,

I ~ " 7 f / ~ ± \ — 7 / 1 I ~ " 7 ( < \ 7 / \ *

n x — ar nl x — ax

Note that the above work is valid whether ax or ar are simple zeros ofgn+1 or not. Also, the first inequality is valid even when r = 2 provided&! is not a simple zero of gn+1, and similarly for the second inequalityprovided ar = α2 is not a simple zero of gn+1.

In order to complete the induction on the middle inequality of (2.9),we suppose first that α2 < s < ar and s ^ x. Taking j = 2 in (2.2) and(2.7), we obtain

I G w + 1 ( # , s) I — { ^ ~ S I GW(OJ, s) I + s ~ α 2 I Gn(a2, s) |

nl ar — aλ

If, however, αx < s <£ α2 and s ^ a;, we again take i = 1, whence

(2.10) I G.+1(», β) I - { J L z A . I G%(X, s) I + 1 ^ 1 ^ | Gn(aus) \\n I x — αx x — &! J

^ 1 1n\ ar — α x

if »! is not a simple zero of gn+u or

I Gn+1(x, s)\=j^ IZl1 ' G*(αi' S)' - i f α l α

if aλ is a simple zero of gn+1. (In this latter case, we used (2.8) andthe first of inequalities (2.9).)

Similarly, if ax < s < αr_x and x ^ s, we take i = r — 1 in (2.7) to obtain

GREEN'S FUNCTION OF AN N-POINT BOUNDARY VALUE PROBLEM 807

I Gn+1(x, β) I ^ i - { S~x I Gu{x, s) I + **-* ~ s I Gn(ar.19 s) \\n I αr_χ — x α r _ x — a? J

n\ ar — aλ

If αr_i S s < ar and x ^ s, we again take j = r, whence

(2.11) I Gn+1(x, s) I ^ — { ^""^ I G Λ (» , s) | + ar ~ s \ Gn(ar, s) \n I α r — x ar — x

^ 1 1

if α r is not a simple zero of gn+lf or

, β) I = i ^ ^ ^ - 1 G.(αr, s) \ ^n ar — x n\ ar — ax

if α r is a simple zero of gn+1.We now complete the induction in the case that gn+1 has only two

distinct zeros. For n ^ 2, at least one of al7 a2 must be a multiple zeroof gn+1. Suppose a2 is a multiple zero of gn+1. Let #w(x, s) denote theGreen's function for the boundary conditions

V(aύ = »'(«!> = = »(fcl)(αi) - 0 ,

( t t l) - 0 ,

and gn+1(x, s) the Green's function for these boundary conditions withk2 replaced by k2 + 1. For any a with ax < a < α2, let 0»+i(fic, s; α) denotethe Green's function for the boundary conditions of gn together withthe condition y(a) — 0. Let Gn(x, s), Gn+1(x, s) Gn+1(x, s; a) denote therelated functions defined by (2.6). By (2.7)

Gn+1(x, s) - i - { x~s Gn(x, s) - ^ J Z l Gn(a2, s)\, x Φ a 2 ,n i x — a2 x — a2 J

Gn+1(x, β; α) = 1 \^—?- Gn(x, s) - ^ - 1 G%(a,n ix — a x — a

For each s φ α2 and a? =£ α2, we have

lim Gn+1(x, s; a) = G»+1(α?, s)

since G%(α, s) is a continuous function of a for any s Φ a2. Since wehave already established

808 PAUL R. BEESACK

,nl a2 — x

\Gn+1(x,s;a)\ ^

1, x <^ s

, αx < s < α2, — co < x < oo ,w! α2 — aτ

1 1 α < s < c o s<xnl x — a±

2 ~~

the same result holds for \Gn+1(x, s)\. The proof of (2.9) is now complete.From (2.6) and (2.9) it follows that, for each n ^ 2, we have

(2.12) \gn(x,s)\ ^ / - ,(n - 1)1 (ar - a,)

We now prove t h a t

(2.13) Π I x - a, |fc*+1 ^i

α,<s<αr, -

n nfor ai^x^ar,

which will complete the proof of (1.4). Here, r ^ 2 , 0^kίf aλ<a2< < α r ,and fci + &2 + + kr + r = n. We note that equality is attained in(2.13) for r = 2, kλ = 0, &2 = n - 2, a? = [(w - l)αx + α2]/^, or for r = 2,&! = ^ — 2, k2 = 0, and a? = [(w — l)α2 + αj/w.

Instead of proving (2.13) in the form stated, we will prove

(2.13,) Π (x - a,) A) 1 (an - axγ

where αx ^ α2 ^ ^ a». As a first step, we prove

for aλ ^ a? ^ α w ,

(2.14) Π (* - at).(a -αO -^α, - *),

To this end, suppose a3 < x < α i + 1 . If cc — αx ^ αw — x, then

Π (» - α4^ (a? ^ (a? - a^~\a% - x)

if x — αx ^ αw — x (and α^ < x < α i + 1 ) , then

Π (x - a,) ^ (x - ax)\an - ^(x- a,) (an -

proving (2.14). Now, setting fλ{x) = (x — αx) (an — #)w~\ we see thatfiix) has an absolute maximum on aλ < x < an when a? — [(^ — l)ax + an]ln.Similarly, f2(x) = (a? — α^^'Xα,, — cc) has an absolute maximum onax < x < an when x — [(n — l)an + a^/n. The inequalities (2.130 and(2.13) now follow by computation.

GREEN'S FUNCTION OF AN JNΓ-POINT BOUNDARY VALUE PROBLEM 809

There remains the question as to whether the above inequalitiesare best possible. The inequality (2.9), or rather the restriction

(2.15) \Gn(x,8)\£- — I -, ax<s<arf - c o <χ< oo ,(n-ΐ)l (ar - α2)

is best possible. Indeed, equality holds in (2.15) for precisely the casesin which equality was attained in (2.13), that is, for r = 2, kx = 0, k2 = n — 1and for r = 2, kx = n — 2, k2 = 0. For the first of these cases we shallshow that

(2.16) Km \Gn(x,a2)\ = 1

(n — 1)! (α2 — αx)

which will prove that (2.15) is best possible. Indeed, taking j = 2 ands = α2 in (2.7), we have

G»+i(α?, α2) = — G%(α?, α2), x Φ a2,

so that (2.16) holds for {n + 1) if it holds for n. One easily verifiesthat (2.16) is valid for n = 2. Similarly, for the second case noted above,we have

lim* - > < * ! - (n — 1)1 (a2 — ax)

It seems likely that equality is possible in (2.15) only in these two cases.Nevertheless, the inequality (1.4) is not the best possible, even in

the simple case r — 2, kx — 0, k2 — 1, when (2.15) is best possible. Weleave it to the reader to verify that in this case

with equality holding for s = 1/2 {(3 - T / Ί Γ K + ("l/ΊΓ - l)α2} = s0, andα? = (a2s0 — af)l(a2 + s0 — 2ax). This is an improvement over our estimate(1.4) which, for this case, is

3* Applications* Consider the ordinary differential equation

(3.1) y { n ) + f(x, y,y',---, y{n~X)) = 0 ,

where we assume that / is a continuous, complex-valued function for

a>i ^ x ^ αr» and for all 2/, /', , 2/(w~~υ, and

810 PAUL R. BEESACK

(3.2) \f(x,y, •• ,» (- 1 )) |^λ(»)l»l

in this domain, where h(x) is a nonnegative continuous function withh(x) ΐ θ on aλ ^ x ^ ar. Suppose (3.1) has a non trivial solution y(x)satisfying the boundary conditions

(3.3) v i a , ) = y ' i d i ) = •• = y ^ f a ) = 0 , l ^ i ^ r ,

where ax < a2 < < ar9 0 ^ &*, fcx + k2 + + kr + r = n. Thenis a solution of the linear nonhomogeneous equation

y™ = -/[&, y(s), ϊ/'(αθ, , ^ - " ( α ) ]

which satisfies the linear homogeneous boundary conditions (3.3). ByTheorem 1 of [2] it follows that y(x) satisfies the integral equation

(3.4) y(x) = \argn{x, s)f[s, y(s), . . , ^ - " ( β f l d β , a, ^ x ^ ar,

where gn(x, s) is the Green's function of the system (1.3). Taking x tobe the point—or one of the points—at which | y(x) \ assumes its maximumvalue on ax ^ x ^ α r, we obtain

(3.5) 1 < \ar\gn(x,s)\h(s)ds,

by (3.2). Hence, using the inequality (1.4),

(3.6) 1 < ( i ^

\ nThe inequality (3.6) is thus a necessary condition for the existence

of a solution of the boundary value problem (3.1), (3.3). If the system(1.3) is self-adjoint, we may improve this necessary condition. The system(1.3) is self-ad joint if n — 2m, r = 2, and the boundary conditions are

ί3 7) \v(μi) = y'{ai) = ' = y { m

The Green's function is now symmetric, and by (2.12) we have

| g . ( * , β)l - \g.(8, x)\ S ^ " ai]l(f2 ~ S)\ , a x < s < a 2 .(2m — 1)! (α2 — α^

On substituting this in (3.5), we obtain

(3.8) 1 < — -f- [\s - aT{a2 - srh(s)ds(2m — 1)! (a2 — aj Uλ

as a necessary condition for the existence of a solution of the boundaryvalue problem consisting of (3.1)—with n = 2m—and (3.7).

GREEN'S FUNCTION OF AN iSΓ-POINT BOUNDARY VALUE PROBLEM 811

We may adopt a different point of view and use (3.6) or (3.8) toobtain an extension of the following oscillation criterion due originallyto Liapounoff (cf. [1]): / / y"(x) and y"{x)y~\x) are continuous foraλ ^ x ^ a2, with y{aλ) — y(a2) = 0, then

(3.9) \°1\y"y-1\ dx >a2 —

By taking / = - y^{x)y-\x)y in (3.1), h(x) = \y(n)(x)y-\x)\ in (3.2), (3.6)leads to the following extension: If y{n){x) and y{n)(x)y~1(x) are continuousfor a1 ^ x g α r, and y(x) has n zeros (counting multiplicity) includinga1 and ar, on ax ^ x ^ α r, then

(3.10) ί"r| dx

This reduces to (3.9) when w = 2. Similarly, using (3.8) in the self-adjointcase: If y{2m)(x) and y{2m)(x)y~1(x) are continuous for aλ ^ x ^ α2,2/(fc)(^i) = 2/(fc)(O = 0 / o r 0 ^ Jfc m - 1, then

(3.11) ( % - αθm(α2 - α?)« |τ/ ( 2 w ) (^- χ (^) | dx > (2m - 1)! (α2 - α j ,

(3.12) (α2| τ / ( «(^-^) I dx > (fm ~ 1 } ' 2 T

The inequality (3.12) also reduces to (3.9) when m = 1, but is betterthan (3.10) for n = 2m ^ 4.

Next we turn to the question of obtaining a lower bound for therath zero of solutions of the linear equation

(3.13) y^ + h(x)y = 0

on an interval /: x0 <£ x < oo. cf. [3, Theorem 5]. We suppose that h(x)is continuous, complex-valued, with h(x) Ξ£ 0 on I, and

(3.14) \~\h(x)\dx = K.

Ifcbi^^i^ ^ dm are m consecutive zeros of any solution of (3.13)on the interval I, then for m^n

(3.15) am > aλ + n - V (ra - ^ + 1) [(^ - 1)!] m

n — 1 f K

To prove this, we first note that for the equation (3.13)—but notnecessarily for (3.1)—no solution can have a zero of multiplicity greaterthan (n — 1) at any point of J. Hence, if a{ 5g ai+1 ^ ^ α ί+%_! aren consecutive zeros of a solution of (3.13) on / t h e n a{ < ai+n-u and (3.6)applies to give

812 PAUL R. BEESACK

(3.16) (—"-JTw! < (ai+m-i - ad"'11^"') h(x) \ dx .

Suppose m = qn + s, where q ^ 1, 0 ^ s < n — 1, so am ^ agn. Takingi = 1, n + 1, , (q — l)w + 1 in (3.16) and adding these inequalities gives

(«) I dx ,- ^ — ) gw! < Σ [din - α ( i _ 1 ) w + 1 ] ^ 1

whence, since qn — m — s ^ m — n + 1,

(3.17) (-±J)n~\m - n + 1)[(n - 1)!] < (am - a^ [m\h(x)\dx .

The inequality (3.15) follows at once from (3.17). As in [3], (3.17) canbe used to obtain a lower bound for am even when K = oo.

In case m > 2n — 1, these inequalities can be improved slightly, asfollows, lί m — qn + s, with tf^l, 0 ^ s ^ n — 1, there exists preciselyone integer r ^ 1 such that

rn — (r — 1) ^ m < (r + l)w — r .

Now taking i = 1, w, 2^ — 1, , (r — l)w — (r — 2) in (3.16), andproceeding as above, we obtain

(—^—V'Vnl < [α_(r_1} - α j - 1 [^^IHx)] dx .

Since r(w — 1) + n > m and αm Ξ> «,.»_(,._!,, we have

n — j

this yields the estimate

(3.18) am > a,n — 1 r (^ — l)K

which is a slight improvement on (3.15) for m > 2n — 1.

REFERENCES

1. G. Borg, O n α Liapounoff criterion of stability, Amer. J. of Math., 7 1 (1949), 67-70.2. K. Toyoda, Green's function in space of one dimension, Tόhoku Math. J., 38 (1933),343-355.3. D. V. V. Wend, On the zeros of solutions of some linear complex differential equations,Pacific J. Math., 1O (1960), 713-722.

CARLETON UNIVERSITY,

OTTAWA, CANADA,

ON MUTOMORPHIC -GROUPS

J. R. BOEN

In a paper to appear, G. Higman has "classified" the finite 2-groupswhose involutions are permuted cyclically by their automorphism groups[1], He found that such a group is either generalized quaternion,abelian of type (2TO, « ,2W), or of exponent four and class two. Healso proved that a finite p-group with an automorphism permuting itssubgroups of order p cyclically is abelian if p is odd. We say that agroup is π-automorphic if it has the property that any two of itselements of order k are conjugate under an automorphism where π isa set of positive integers and keπ. In this paper we conjucture thata finite p-automorphic p-group is abelian for odd p, and prove that acounterexample cannot be generated by fewer than four elements.

We use the following notation. Let pn+1 be the exponent of theP'gYonp G; Hk(G) denotes the set of elements of G whose orders do notexceed pk; G' is the commutator subgroup of G; (x, y) — x~Ύy~xxy\ Z(G)is the center of G and Z2(G) is the preimage of Z(G[Z) in the cannonicalhomomorphism of G onto G/Z; Φ(H) is the Frattini subgroup of thegroup H; \H\ is the order of H; | x | is the order of the element x.GL(3, p) is the full linear group of degree three over the prime Galoisfield GF(p).

Henceforth let G denote a finite p-automorphic non-abelian p-groupfor odd p. Note that Hλ{G) = H,QZ = Z(G), so H, is a subgroup.

LEMMA 1. GjHx is p-automorphic.

Proof. Clearly there exists x e Z2(G) such that | x | = p2 because Gcannot be of exponent p. Consider yeG where | y | == p2. By thedefinition of G there exists a e Aut (G) such that (y*)" = xv. Let y* =

wx. Thus (y")p = (wx)p = wpxp(x, wy2' by the choice of x. If Z has anelement of order p2, choose x to be it. Then (x, w) — 1. If Z = Hlf

then (x, w) e Hx and (x, wy2) — 1. In either case (ya)p ~ (yp)a = x* =wpxp so weH, and (yHJ* = xHτ. Q.E.D.

LEMMA 2. / / G' = fli, ίfce^ Hn(G) = Φ(G) = Z.

Proof. Φ(G) — Φ — G'P where P is the subgroup of G generatedby pth powers. Gr = Hλ implies that G is of class two, so (xp, y) =(a?, yp) = (a?, 1/)2> = 1. Hence Φ ξ^Z. In the canonical homomorphism of (?

Received March 31, 1961, and in revised form August 31, 1961. This paper was spon-sored in part by NSF Grant G-9504.

813

814 J. R. BOEN

onto G\G' = K, Hn(G) = Hn is the preimage of Hn^(K) = Φ(K). (Hn isa subgroup because G is regular; K is abelian and has equal invariants).If there exists xe Z such that | x | = p%+1 then for any yeG where|t/ | = p%+1

W e have (#**)* = x*n for some <xeAut(G). By the samereasoning used in Lemma 1 it follows that y* = wx where we Hn.Hence yaeZ so yeZ and G is abelian, a contradiction. Q.E.D.

LEMMA 3. If Gf — Hu then φ:x—>xpn is an isomorphism of G[Zonto G\

Proof. Since G is of class two, (xy)m = xmym(y, xy2 ' where m — pn.fm\ (v)

But ô ) is a multiple of p so {y, x)X2' = 1 and φ is an endomorphismof G. Clearly Hn = Z is the kernel of φ. At least one nonidentityelement of G' is an mth power, hence every one is and thus G\Z = Gr.Q.E.D.

THEOREM. A finite non-άbelian p-automorphic p-group G cannotbe generated by fewer than four elements.

Proof. It is easily seen that Ή^^Φ. By repeated application ofLemma 1 we arrive at a Gλ such that G\ = Hλ{G^) where Gλ has the samenumber of generators as G. Since we argue by contradiction we mayassume without loss of generality that Gr — Hλ.

Clearly G cannot be cyclic. If G can be generated by two elements,the fact that G is of class two implies that G' is cyclic; this contradictsLemma 3. Hence we assume G to be a three-generator group, sayG = {uu u2j u3}. Lemma 2 implies the following identities.

( i ) (upupuph, uûψuψh') = Π < < i * i 5 l ' ' " " a w w h e r e h,h'eZ a n d si3 = (ui9 u3).

(ii) (u?u?uψhγn = Π*?' where t, = uf.

Now every element of G' is a commutator. Thus there exist rela-tions ti = Siί' iί Sa?'8, i = 1, 2, 3, where | A | = | (α^ ) | =£ 0. Let α be anautomorphism of G, say u* = ulûlûl^hi, i — 1, 2, 3, where ^ G ^ andxί3eGF(p). (i) implies that s?y = Π*<ΪS*?*1 where »Aι = a^α^ - ^ fexα.Hence

But (ii) implies that

*? =

Equating these two representations of tf and noting that s12, s13, and s2

are independent, we have

ON p-AUTOMORPHIC p-GROUPS 815

(iii) AX = XA

where A — (αί7 ), X — {xi0)y and X = (a?, , ) are nonsingular 3-square ma-trices over GF(p). It is clear that X=\X\B~1X-TB where X~τ isthe transpose of X~x and B — {bi3) has the entries δ13 = δ31 = — δ22 = 1and the remaining δ^ = 0. Thus, substituting for X in (iii), we equatethe determinants of the two sides of (iii) and find that | X | = 1. (iii)then takes the form:

(iv) CX^C-1 = X where C = AB'1 .

It follows that (iv) holds for all X in some transitive (on the non-zero vectors of the 3-space V) subgroup T of GL(3, p). Thus | T | isdivisible by p* - 1. | GL(3, p) | = p3(^ - l)(p2 - 1)(^3 - 1). Let q be aprime divisor of p2 + p + 1 where tf > 3. It is easily shown that sucha tf exists and that q is relatively prime to p — 1 and p + 1. Thus aSylow g-subgroup of T is a Sylow g-subgroup GL(3, p). GL(3, p) con-tains a cyclic transitive subgroup of order j>8 — 1, the multiplicativegroup of the right-regular representation of GF(pz) considered as avector space over GF(p). Hence a Sylow g-subgroup of GL(3, p) iscyclic, so an Xe T of order q is conjugate to

(ω \Y —\ ωp where co" = 1

in GL(3, p3). But F is certainly not conjugate to Y~τ in GL(3, p3)from which it follows that X will not satisfy (iv), a contradiction.Q.E.D.

The author is indebted to G. Higman and G. E. Wall for theirsuggestions, and to the referee for correcting an error.

REFERENCE

1. G. Higman, Suzuki 2-groups, to appear.

UNIVERSITY OF CHICAGO

FURTHER RESULTS ON -AUTOMORPHIC ^-GROUPS

J. BόEN, 0 . ROTHAUS, AND J. THOMPSON

Graham Hίgman [3] has shown that a finite p-group, p an odd prime,with an automorphism permuting the subgroups of order p cyclically isabelian. In [1] a p-group was defined to be p-automorphic if its auto-morphism group is transitive on the elements of order p. It was con-jectured that a p-automorphic p-group (p Φ 2) is abelian and proved thata counterexample must be generated by at least four elements. In thispresent paper we prove that a counterexample generated by n elementsmust be such that n > 5 and, if n Φ 6, then p < w3** (Theorem 3). Wealso show that the existence of a counterexample implies the existenceof a certain algebraic configuration (Theorem 1). All groups consideredare finite.

Notation. Φ(P) is the Frattini subgroup of the p-group P and Pr

is its commutator subgroup. £?;(P) is the subgroup generated by theelements of P whose orders do not exceed p\ Z(P) is the center of P.F(mf n, p) denotes the set of p-automorphίc p-groups P which enjoy theadditional properties:

1. P' — f?i(P) is elementary abelian of order pn.2. Φ(P) = Z(P) = Ωm(P) is the direct product of n cyclic groups

of order pm.3. | P : 0 ( P ) | = J> .In [1] it was shown that a counterexample generated by n elements

has a quotient group in F(m, n, p). Hence, in arguing by contradiction,we may assume that a counterexample P is in F(mf n, p).

Let A = A(P) = Aut P and let Ao = ker(Aut P-> Aut PjΦ{P)). ThusAIA0 = B is faithfully represented as linear transformations of V— PJΦ(P),considered as a vector space over GF(p).

Since p is odd and d(P) = 2, the mapping rj:x—>xpm is an endo-morphism of P which commutes with each σ of Aut P. Since Ωm{P) —Φ(P), ker η = Φ(P), so rj induces an isomorphism of V into W = Pr.Since dim V = dim W, Ύ] is onto.

The commutator function induces a skew-symmetric bilinear map ofV x V onto W, (onto since P is p-automorphic) and since Φ(P) = (P),(,) is nondegenerate. Associated with (,) is a nonassociative producto, defined as follows: Ifa,βe V, say a = xΦ{P), β — yΦ(P), then [x, y]is an element of W which depends only on a, β, and so [x, y] = zpm wherethe coset y = zΦ(P) depends only on a, β. We write aoβ = y. Animmediate consequence of this condition is the statement that a —• aoβ

Received December 6, 1961. Boen's work on this paper was partly supported by N.S.FGrant G-9504.

817

818 J. BOEN, O. ROTHAUS, AND J. THOMPSON

is a linear map φβ of V into F. Thus, o induces a map Θ of V intoEnd F, the ring of linear transformations of V to F.

If σ is the inner automorphism of End V induced by σ e B, then thediagram

V-?-+ End V

lF — End V

commutes, that is φβσ = σ^φβO. Since P is p-automorphic, if α, /3 arenonzero elements of F, then a = /3σ for suitable σ e B, so that ^Λ = σ~V

THEOREM 1. J/ α e F, ίfeβw ^ is nίlpotent.

Proof. We can suppose a =£ 0. Since tfoα: = 0, φΛ is singular. Letf(x) = xn + c1^

n"1 + caίcn""a + be the characteristic equation of φa.

f(x) is independent of the nonzero element a of F, and cn — 0 since0α> is singular.

Let ffi, * , α n be a basis for F, and identify ^ with the matrixwhich is associated with φΛ and the basis alf , an. Then c< is the sumof all i by i principal minors of φa, so if a — X^ + + \nocnj Cι is ahomogeneous polynomial of degree ΐ(ίg w — 1) in the w variables λx, ,λn. By a Theorem of Chevalley [2], there are values λ1? , λΛ of GF(p)which are not all zero, such that c{ = 0. Since c* is independent of thenon-zero tuple (\19

β ,λn), it follows that c< = 0 so ^Λ is nilpotent.Theorem 1 states that θ( V) is a linear variety of End (F) consisting

only of nilpotent matrices such that any two nonzero x, yeθ(V) aresimilar. If one could show that the algebra generated by Θ(V) werenilpotent, an easy argument would show that all p-automorphic p-groups(p odd) are abelian.

THEOREM 2. Let r be the rank of φΛ. Ifn>3, then 2 < r < n — 1.

Proof. We assume n > 3 because n 5g 3 was treated in [1], Clearlyr Φ 0 because P is non-abelian and r Φ n by Theorem 1.

Case /. r Φ n — 1. Suppose r = n — 1. Then, for a: =£ 0, βoa =βφa — 0 implies that β e {a} where {a} is the subspace of F spanned bya. If yφl = (7φa)φa = 0, then T^^ e {α}, say γ^,, = to. But 7^* + < v =0 by the skew-symmetry of o, so aφy = —to. By Theorem 1, fe = 0 andthus je{a}. Hence rank φl = rank 0Λ, a contradiction to Theorem 1.

Case II. r Φ 1. Choose a basis of F, say α l f •••,«., and suppose

FURTHER RESULTS ON p-AUTOMORPHIC p-GROUPS 819

that φa = {da) with respect to this basis; End (F) has the obvious matrixrepresentation with φae θ(V)a End(F). Recall that Θ(V) becomes anw-space of n by n nilpotent matrices over GF(p) in which any twononzero matrices are similar. If r = 1, then we may assume withoutloss of generality that φΛ has a 1 in the (1, 2) position and zeros else-where.

If every (xiS) = Xeθ(V) satisfies xi3 = 0 for i > 1, then we are donebecause the nilpotency of Ximplies that xn = 0 for every J e Θ(V), whichimplies that dim#(F) < n. If, on the other hand, there exists Xeθ(V)with a nonzero entry below the first row, then we may use the factthat every 2 by 2 subdeterminant of every element of θ( V) vanishes toshow that every X has its nonzero elements in the second column only.But the nilpotency of X implies that x22 = 0. Hence dim θ( V) < n, acontradiction.

Case III. r Φ 2. // r = 2, we may assume without loss of gener-ality that

(a) φa has Γs in the (1, 2), (2, 3) positions and zeros elsewhere orelse

(b) φa has Γs in the (1, 2), (3, 4) positions and zeros elsewhere.First consider (a).

If every (xiS) = Xe 0(F) satisfies xi5 = 0 for i > 2, then Z{P) £ Φ(P),a contradiction. If every Xeθ(V) satisfies xiS = 0 for j Φ 2, 3, thenίc32 = o because X + kφa is nilpotant for every k e GF(p) and p > 2. Butthen dim#(F) < n, a contradiction. Hence we need consider only thesubcase of (a) in which some Xeθ(V) has a nonzero entry below thethird row and a nonzero entry that is not in columns two or three. Con-sider such an X Unless xiS = 0 when i Ψ 1, 2 and j φ 2, 3, it is easyto see that there exists a nonzero 3 by 3 determinant in X + kφa forsome k. It is also easy to see that any two rows of X below the secondrow are dependent, and that any two columns other than the secondand third are dependent. Using the fact that every 3 by 3 subdeter-minant of every element of Θ(V) is zero, it is straightforward to showthat there exist nonsingular matrices R and S such that RXS has Γsin the (1, 4), (3, 2) posititions and zeroes elsewhere and RφaS has Γs inthe (1, 3), (2, 2) positions and zeroes elsewhere.

Set X' = -RXS, φf

a = RφooS. It is now straightforward to show thatthat if Y=(yij)eBΘ(V)S is linearly independent from {X',φ'a}f thenyiά = 0 for i Φ 1 and j φ 2. This implies that dim Rθ{ V)S < n, a con-tradiction, since dim Rθ( V)S = dim θ( V) = n.

Subcase (b), in which φ\ — 0, is handled in a similar fashion exceptthat we exclude the case in which every Xeθ(V) satisfies xi5 = Q, j Φ 2,4,by noting the following: In such a case (X + kφaf = 0 for every k impliesthat x22 = 0, which in turn implies that dim#(F) < n.

820 J. BOEN, O. ROTHAUS, AND J. THOMPSON

COROLLARY. F(m, n, p) is empty for all m and odd p unless n > 5.

Proof. Theorem 2 implies that n > 4 and that if n = 5, then rankφa = 3. Let Sn denote the projective (n — l)-space whose points arethe 1-subspaces of 7. If n = 5 and rank φa = 3, then it follows thatSδ is partitioned into lines according to the rule that {α}, {β} (0 Φ a,βe V) lie on the same line if and only if aoβ = 0. But S5 hasP4 + Pz + p2 + P + 1 points and cannot be partitioned into disjoint subsetsof p + 1 points each.

THEOREM 3. If p^ nSn2 and n Φ 6, then F{m, n, p) is empty forall positive inteqers m.

Proof. If GL (n, p) denotes the invertible elements of End V, then

I GL(n, p) I = pn{n-1)/2-k(n, p), where k(n, p) - (p* - I X P * " 1 - 1) (p - 1).

If we consider GF{pn) as a vector space over GF{p), the right-regularrepresentation shows that GL(n, p) contains a cyclic group of order pn — 1.

Let Φd(x) be the monic polynomial whose complex roots are the primi-tive dth roots of unity. Then pn — 1 — ϊ[d\nΦd(p). By an elementarynumber-theoretic theorem [4], Φn(p) and k(n, p)jΦn(p) are relatively prime,or their greatest common divisor is q where q is the largest prime divisorof n, in which case Φn(p)lq is relatively prime to k(n, p)l<Pn(p) Thus,we determine ε — 0 or 1 so that Φn(p)/qε is relatively prime to k(p9n)/Φn(p).

Let peF(m,n, p). Since P is p-automorphic, \B\ is divisible bypn — 1 and in particular is divisible by Φn(p)/qs. Let r" be the largestpower of the prime r which divides Φn(p)lQ2, a ^ 1> and let Sr be a Sylowr-subgroup of B. By Sylow's theorem and the preceding paragraph, Sr

is cyclic with generator σr.

Since P belongs to the exponent n modulo r, it follows that λ, Xp,• , λ2^"1 are the characteristic roots of σr, λ being a primitive r*th rootof unity in GF(pn).

Since η commutes with σr, λ is also a characteristic root of σr onW. Since (α, β)σ — {μσ, βσ), the characteristic roots of σr on W are tobe found among the χpi+pj

9 0^i<j^n — 1, as can be seen by diago-nalizing σr over V(&GF{pn). Hence, λ = χpi+pj for suitable i,j and so

(1) p* + pj - 1 = 0 (mod rΛ) .

Since r was any prime divisor of Φn(p)lq2, we have

(2) Π (P* + P'" - 1) = 0 (mod Φn{pW)

The polynomials Φn(x), n Φ 6, and α* + x j - 1 are relatively prime, a fact

FURTHER RESULTS ON p-AUTOMORPHIC p-GROUPS 821

which can be seen geometrically, as pointed out by G. Higman. Name-ly, if ε, e' are complex numbers of absolute value one, and ε + ε' = 1,then the points 0,1, ε are the vertices of an equilateral triangle, so thatε is a primitive sixth root of unity. Since n Φ 6, we can therefore findintegral polynomials fix), g{x), such that

(3) f(x)Φn(x) + 9(x) Π (*' + x* - 1) = INI ,

where

(4) N

is the resultant of Φn{x) and Πfa* + xj — 1).From (4) we see that N ^ sφ{n)n2, since there are at most φ(n)n2 triples

(?, i,j). Now (2) and (3), the fact that Φn(p)/qζ divides | JV|, imply that

(5) ΦAVW ^ &{n)n2 .

One sees geometrically that ΦJj>) (p — l)φ{n), so with (5) and qs ^ nwe find

(6) p ^ 1 + nllφM3n2 < n3n2 .

REMARK. Theorem 3 of [3] provides a certain motivation for thedetailed examination of Φn(p) in the preceding theorem.

BIBLIOGRAPHY

1. J. Boen, On p-Automorphic p-Groups, (to appear in Pacific Journal of Mathematics).2. C. Chevalley, Demonstration d'une hypothese de M. Artίn, Abh. Math. Seminar U.Hamburg, 11 (1936).3. G. Higman, Suzuki 2-groups, (to appear).4. T. Nagell, Introduction to Elementary Number Theory, Wiley (1951).

UNIVERSITY OF CHICAGO AND UNIVERSITY OF MICHIGAN

INSTITUTE FOR DEFENSE ANALYSES

UNIVERSITY OF CHICAGO

BOUNDS IN THE NEUMANN PROBLEM FOR SECOND

ORDER UNIFORMLY ELLIPTIC OPERATORS

J. H. BRAMBLE AND L. E. PAYNE

l Introduction. In this paper we derive certain a priori in-equalities which are useful for obtaining bounds in the interior'Neumannproblem for second order elliptic partial differential equations. In es-tablishing these inequalities by our methods it is necessary to obtainlower bounds for the inverse of the Poincare constant {μ2 of eq. 3.3)and the first nonzero Steklov eigenvalue (p2 of eq. 3.11). An optimalPoincare inequality for convex domains in ^-dimensions was given byPayne and Weinberger [5], and a method for obtaining lower bounds forp2 for ^-dimensional star-shaped regions was indicated by Payne andWeinberger [3]. However, to the authors' knowledge, no explicit lowerbounds for p2 and μ2 for general w-dimensional regions have previouslybeen given. Lower bounds for these constants which lead to the abovementioned inequalities in the Neumann problem are of interest in them-selves and should prove useful in other applications.

For the special case of the Laplace equation other methods forderiving bounds for the Dirichlet integral in the Neumann problemappear in the literature (see [2], [6]). For starshaped regions a methodsimilar to that proposed here was obtained in [3]. Bounds in exteriorNeumann problems were given in [4].

2* Preliminary inequalities. Let R be a simply connected boundedregion with boundary C in Euclidean w-space. In R we assume thatthe operator L given by

(2.1) Lu = (ai3'uti)j

is a uniformly elliptic operator, defined for sufficiently smooth functionsu. In (2.1) ,i denotes partial differentiation with respect to the coordi-nate xi and the summation convention is assumed. The coefficientmatrix aij is symmetric and the condition of uniform ellipticity may bestated as follows: There exist positive constants a0 and ax such thatfor every real vector (ξu , ξn), the relation

(2.2)

is valid uniformly in R.

Received December 4, 1961. This research was supported in part by the United StatesAir Force through the Air Force Office of Scientific Research of the Air Resarch and De-velopment Command under Contract No. AF 49 (638) 228.

823

824 J. H. BRAMBLE AND L. E. PAYNE

We consider now an arbitrary point of R which we choose to bethe origin. Let Sa be the interior of a sphere of radius α, with centerat the origin and such that Sa c R. The surface of the sphere will becalled Σa. We denote by Ra the region R — Sa, where Sa is the closureof Sa.

Let u be any sufficiently smooth function in R + C and let /* bea sufficiently smooth vector field defined in Ra. Then, by the divergencetheorem, we have

(2.3) £ f'UiU'ds = - £ f'n^ds + \ fUu2dv + 2ί f*uu9idv ,J 0 J Σa jEa jRa

where n{ is the component of the unit normal directed outward on C.An application of the arithmetic-geometric mean inequality applied tothe last term on the right of (2.3) yields

(2.4) <f fn^ds ^ - <ί pn^ds + f (f\ + l fψλtfdv + [ an tu {dv

where a is some positive function in Ra.We assume now that fι and a have been chosen so that

t = f% ^ Kx > 0 on C

(2.5) - f% ^ K2 on Σa

fU + \fΨ ^ 0 in Ra,

where Kx and K2 are constants. (We shall in a subsequent sectionconstruct vectors fι satisfying (2.5) for certain domains.) Using con-ditions (2.5) with (2.4) we have that

(2.6) <ί tu2ds ^ K2 <f u2ds + a[ u tu {dv ,JO J Σa jRa

where a is an upper bound for a in Ra. Suppose that u is normalizedsuch that

(2.7)

Then

(2.8) φ u2ds ^ pΛ u tu idvJ Σa JSa

where p2 is the first nonzero eigenvalue in the Steklov problem for thesphere Sa. That is

BOUNDS IN THE NEUMANN PROBLEM 825

(2.9) p2 = min= TYΊ1Π i£α£

fwhere the minimum is taken over all sufficiently smooth functions inSa which satisfy (2.7). For the sphere of radius α, p2 is explicitlygiven by

(2.10) p2 = 1/α .

Combining (2.6) and (2.8) it follows that

(2.11) <f tu2ds ^ KdD(u, u) = KS U tutidv ,JO JB

where K3 = max (aK2, a), or using (2.5)

(2.12) <f u*ds (KJKJ D{u, u) .

Now from the divergence theorem

(2.13) f x%u2ds = nί ΐΛfo; + 2ί xluu {dv .

Using the arithme trie-geometric mean inequality it follows easily that

(2.14) u'dv ^ =s- φ u'ds + ^-D(u, u) ,

where ru is the maximum distance from the origin to C. Inequality(2.12) with (2.14) yields

(2.15) ί u'dv g KJ){μ, u) ,

J R

where

9/v r 2r

Q + —n

The preceding inequalities depended entirely on the existence of avector field /•' satisfying (2.5). In certain cases, as will be shown in asubsequent section, such a vector field can be explicitly constructed soas to yield explicit, easily computable constants Ku K2 and iζ$.

In some cases it may be that for the region R the vector field fι

is quite difficult to construct. We can make use of an additional in-equality to reduce the problem to that of obtaining an inequality ofthe form (2.12) for a subregion of R.

Let us divide the region R into two disjoint subregions Rx and R2.

These regions are to be separated by a surface C\ The portion Cwhich is part of the boundary of i2< will be denoted by Ci9 i — 1, 2.Thus the boundary of R{ will be Ct + C. We further assume that thesubdivision has been made in such a way that CΊ is star shaped withrespect to some point P not in Rλ + C". We choose P to be the originand apply the divergence theorem is Rλ to obtain

(2.16) \ x'r-^n^ds = -\ r-{n+ )u2dv

for any function u sufficiently smooth in R. Defining i = XiΠjr andusing the arithmetic-geometric mean inequality we obtain

(2.17) <f u2ds JL(I*.Y ( u2ds + ISJΣMXD^U, U) .

In (2.17) rM and rm denote upper and lower bounds for r in Rλ and tm

a lower bound for t on Cx Dλ{u, u) denotes the Dirichlet integralover Rlm

Now suppose that for R2 we could find a vector field fι satisfying(2.5) relative to R2 and obtain the inequality

(2.18) <f u2ds ^ KJK, D2(u, u) .J c2+c

Then clearly (2.17), together with (2.18) would yield

(2.19) (f u2ds ^ K6D(u, u)J

where of course u is assumed normalized with respect to Sa in R2, andK6 is a constant.

It is now obvious that such a procedure could be repeated a finitenumber of times, finally reducing the region to one for which the in-equality (2.12) may be more easily obtained. In particular if we iteratethis procedure until the gth region Rq is star shaped, then, as we shallsee in § 4, a vector field fι for Rq is easily constructed.

3. Lower bounds for eigenvalues* The first nonzero eigenvalue fi3

in the free membrane problem for R satisfies

(3.1) Δv

and

(3.2)dn

where v is the corresponding eigenfunction. It is well known that μ2


may be characterized by the minimum principle

(3.3) /*2

φ2dvR

for sufficiently smooth functions φ satisfying

(3.4)

and that v is the minimizing function. That is

(3.5) ψ*±v2dv

JR

Now let u = v + Cj where

(3.6) C l = * — ί vdsωna

n~x ha

ωn denoting the surface area of the %-dimensional unit sphere. Thenu satisfies

(3.7) ( u ds = 0 ,

and hence by (2.15)

(3.8) f u2dv ^ KJ)(u, u) = K,D(vf v) .

But

(3.9) ( u2dv = ( v2dv + c\\ dv^\ v2dv .JR JR JR JR

Thus

(3.10) ±. £ ψ^l

or l/if4 is a lower bound for μ2.A lower bound is also easily obtained for p2, the first nonzero

eigenvalue in the Steklov problem for R. Let w be the correspondingeigenfunction. Then we have that

(3.11) v, = n{W'W)

f w2dso

and

(3.12) <f wds = 0 .J o

Now let u = w + c2 where

(3.13) c2 = ί wds .ωan-χ J

Then

(3.14) <f mis = 0

and we may apply (2.12) to u. But by (3.14) we have that

(3.15) <f u2ds = f w2ds + cl ί ds <ί w2ds .Jo Jo Jo Jo

Thus from (2.12) and (3.15)

KJK. £ -f^L = ft f

J c

which gives the desired lower bound for p2.

4. Bounds in the Neumann problem for L. We assume now thatψ is any sufficiently smooth function in R + C. We shall obtain boundsfor the generalized Dirichlet integral, A(φ, ψ), given by

(4.1) A(ψ, ψ) = ί aijf ^ jdv

in terms of Lψ in R and

(4.2) ^ - = a^Utψ j on C .

We take w = ^ + c3 where

(4.3) c3 = — - f ψds .

As before

(4.4) f uds = 0 ,

Now by Green's identity


(4.5) A(u, u) = I u-^—ds — I uLψdv .

We have used the fact that u and ψ differ only by a constant. BySchwarz's inequality we have that

Because of (4.4) inequalities (2.11) and (2.15) are applicable and weobtain that

(4.7) A(f, Ψr * (^T( f t-α0

since

D(u, u) ^ —A(u, u) = —

The inequalities of this section and §2 together with a mean valueinequality given in [1] give immediately interior pointwise bounds for

Ψ + C3.As an application of the results of this section we note here that

(4.7) may be used in conjunction with the Rayleigh-Ritz procedure toyield close bounds for the Dirichlet integral in a specific Neumannproblem c.f. [1].

5. Construction of the vector field. We shall show in some caseshow to construct a vector field satisfying (2.5).

(a) Star shaped regions.We consider the case where C is star shaped with respect to some

point. We choose this point to be the origin. Then if we take

(5.1) / * = χίr-{n+1)

and

(5.2) a = r" ( w - υ .

We have that

(5.3) t = f% = £%,r- ( % + 1 ) ^ hmrΰ{n+1) on C

where h(P) is the distance from the origin to the tangent plane at apoint P on C and hm is the minimum of this function. The condition

830 j . H. BRAMBLE AND L. E. PAYNE

of star-shapedness insures that hm > 0. Since n{ = — a?*/α on Σa itfollows that

(5.4) -/% = a~n on J β

and

fU + j-fψ = 0 in Ra .

In this case, taking a = rm, we obtain

(5.5) ft ^

L\r w / hm nλ

and

(5.6) ^

A different method for obtaining a lower bound for p2 for star shapedregions has been indicated by Payne and Weinberger [3]. For convexregion Payne and Weinberger [5] also obtained the optimal lower boundμ2 ^ π2d~2 where d is the diameter of R.(b) Smooth boundaries.

Let R be a region whose boundary C has continuous curvature.Call the largest principal curvature at a point P of C, KM(p). Let ρ(p)be the radius of a sphere which is tangent to C at P and contained inR. In addition we require p(p) to be less than KM{p)~ι. Denote by Ka bound for the maximum of p(p)~λ for PeC. We consider the familyof parallel surfaces

(5.7) N(x) = JVXα?1, , xn) = constant

with C given by

(5.8) tf(α) = 0

and

(5.9) 0 ^ iSΓ(α?) :S 1/Jf .

The outward normal vector % is defined in this strip and satisfies

(5.10) niti{x) = J(x)

where J(x) is the average curvature of the surface given by N(x) atthe point x — (x1, •••, xn) c.f. [7, p. 3]. We assume also that K ischosen so that

(5.11) J(x)^K.

The above conditions and definitions involve the smoothness of C andessentially the thickness of R. We impose a further condition on theshape of R.

We assume that there is a point, which we choose to be the origin,such that

(5.12) ^ ^ ^ -v + β > -P > - 1r

for some constants p and β > 0 in the strip 0 rg N(x) ^ IJK. In thiscase /* may be chosen as

([vnil - KN(x)) + x'lφ-% 0 ^ iSΓ(a?) ^ I/if

<5'13) ^ = 1 J ^ 9 otherwise( r

with # to be determined. Condition (5.12) means that there is an opensubset Ω of R which has the property that no ray from the boundaryin the direction of the outward normal intersects Ω.

Let a now be chosen so that Sa does not intersect the boundarystrip.

Now on C

(5.14) /%, = [p + - ^

For 0 S N(x) IIK

(5.15) /% - | p [ J ( l - KN(x)) + K]r + n - l - q

Γ^i^(l _ KN{x))

since nld\dx%)N ^ — 1. Because of (5.12) and the fact that 0 ^KN(x) ^ 1 we have that

(5.16) fU ύ {2KrM + n - 1 - q(l -

Now if we choose

(5.17) q =1 — v

it follows that

(5.18) fU ^ -4r~ ( g + 1 ) ^ - 4 r ^ (

in the boundary strip. In the remaining part of Ra we have, since

q ^ n + 3

(5.19) /*< = [n~-(q + l)]r~iq+1) g - 4 ?

Now choose a = r w . Then

(5.20) fU + 1/y* g O i n i ί . .

On Σa

(5.21) - / % , = a'9 .

In this case we have

(5.22) μ2 ^

and

(5.23)

(c) Boundaries with star-shaped irregularities.Suppose now that the boundary C consists of two parts d and C2

where CΊ is smooth and C2 is star-shaped with respect to the chosen origin.We assume that the closure of the interior of C2 contains C2. Forexample, in two dimensions, the components of C2 cannot be isolatedpoints. Let K now be defined relative to CΊ. Denote by R± the regionconsisting of R minus the strip adjacent to d. We suppose that K islarge enough (the strip small enough) to make R1 connected.

We assume that β is such that on C2

(5.24) J ^ ί - ^ β .

Then in place of (5.13) we have

[pm(l - KN(x)) + —V*, in R - J^r J

—r~q , in Rx.r

(5.25) / ' =

Since in the identity (2.3) it is only necessary that /* have a continuousnormal component on the boundaries of subregions of Ra this definitionof f* has sufficient smoothness properties.

In this case we again have the inequalities (5.22) and (5.23),

BOUNDS IN T H E NEUMANN PROBLEM 833

BIBLIOGRAPHY

1. J. H. Bramble and L. E. Payne, Bounds for solutions of second order elliptic partialdifferential equations, Contributions to Diff. Eqtns. (to appear).2. J. B. Diaz and A. Weinstein, Schwarz' inequality and the methods of Rayleigh-Ritzand Trefftz, J. Math, and Physics, 26 (1947), 133-136: see also J. B., Diaz, Upper andlower bounds for quadratic integrals, Collectanea Mathematica, 4 (1951), 3-50.3. L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems,J. Math. Phys. (1955), 291-307.4. , New bounds for solutions of second order elliptic partial differential equations,Pacific. J. Math., 8 (1958), 551-573.5. , An optimal Poincare inequality for convex domains, Arch. Rat. Mech. Analy.,5 (1960), 280-292.6. J. L. Synge, The method of hypercircle in function-space for boundary value problems,Proc. Roy. Soc. London, A 191 (1947), 447-467; se also J. L. Synge, The hypercircle inmathematical physics, Cambridge Univ. Press (1957).7. C. E. Weatherburn, Differential geometry in three dimensions, Cambridge Univ. Press,vol. II (1930).

INSTITUTE FOR FLUID DYNAMICS AND APPLIED MATHEMATICS

UNIVERSITY OF MARYLAND

COLLEGE PARK, MARYLAND

APPLICATIONS OF ULTRAPRODUCTS OF PAIRS

OF CARDINALS TO THE THEORY OF MODELS

C. C. CHANG AND H. JEROME KEISLER

Introduction. Recently in [8] Vaught introduced the interestingnotion of a pair of cardinals (tc(A), tc(R0)) for a model <A, Ro, •> of agiven first-order theory with identity. He proved that if a theory (withcountably many nonlogical constants) has a model with a pair of cardi-nals {a, β) where ω g β < α, then it has a model with the pair ofcardinals (ωl9 ω0). In this paper we have obtained a number of resultsalong the same lines (they may be found in detail in §4); roughlyspeaking, our results are concerned with increasing one or both of thecardinals in the pair {a, β).

It turns out that most of our results on pairs of cardinals of amodel are simple consequences of set-theoretical theorems concerningultraproducts, ultrapowers, and limit ultrapowers of pairs of cardinals.We have isolated these set-theoretical theorems in § 2 and § 3, wherethey are presented with no reference to model theory.

In the last section of the paper, we give some counterexamples tocertain plausible conjectures analogous to Vaught's and our results. Weconclude the paper by stating a number of open problems. We wishto make it clear here that we do not claim to have originated all ofthese problems; in view of Vaught's result, some of them arise quitenaturally and undoubtedly have been considered before.

1. Preliminaries* We employ the usual symbols ε, g , Π, U, Π>U, to denote the various familiar set-theoretical notions. The expression{t I φ(t)} shall denote the set of all elements t such that φ(t) holds.Ordinal numbers will be denoted by ξ, ξ, η, and natural numbers (finiteordinal numbers) by m, n, p. The symbols 0,1, 2, •••, denote the firstnatural numbers. We suppose the ordinals have been defined so thateach ordinal coincides with the set of all smaller ordinals. Thus inparticular 0 is the empty set. We identify cardinal numbers with thecorresponding initial ordinal numbers. The letters a, β, y, 8, denotearbitrary cardinals; ω denotes the smallest infinite cardinal; a+ denotesthe smallest cardinal greater than a. For each ordinal ξ, ωζ denotesthe smallest infinite cardinal which exceeds ωζ for each ξ < ξ. Thecofinality of the cardinal a is denoted by cf(a). (See [6] for its definition

Received November 6, 1961. The results of this paper have been previously announcedin [1]. The authors became interested in this problem through R. L. Vaught. The prepa-ration of the paper was supported in the case of the first-named author by the NationalScience Foundation under grant G-14092.

835

836 C. C. CHANG AND H. JEROME KEISLER

and elementary properties.) The notion of the sum ξ + ξ of two ordinalsI, ξ is assumed to be known. Let a{ be a cardinal for each i e I; Σie/^iand ΠieA denote, respectively, their cardinal sum and cardinal product.aβ shall denote the cardinal a to the power β; a- shall denote the cardi-nal a to the weak power β, i.e., Σy<j8tf

v. Let X, I, and X{ for eachi e I be arbitrary sets. ^ e j^Q denotes the cartesian product of thesets Xi with iel, and Λ:(X) denotes the cardinal of X. We assumethe reader is familiar with the notions of a filter on /, and an ultrafilteron J. Let D be an ultrafilter on I. For any functions f,ge &\eiaiiwe write f =Dg (read / and g are equivalent modulo D) if {i e I\f(ί) =g(i)}eD. The statement / =Dg has the intuitive meaning that / andg are equal almost everywhere. It is proved in [2] that =D is anequivalence relation on teiaiΛ For each fe&*eiaif let f\Ό = {g\g =D/},the equivalence class of / with respect to D. By the (cardinal) ultra-product of the cardinals ai modulo D, in symbols, ILeΛ/Ά w e meanthe cardinal κ{flD\fe^ϊei(χi} By the (cardinal) ultrapower of thecardinal a modulo D, we mean the cardinal a1 ID = ILe/α/^ β is saidto be a (cardinal) ξ-limit ultrapower of a if there exist functions 7,J, ί? with domain f + 1 such that the following hold:

( i ) a = Ύo;

( ϋ ) β = Ύξ;

(iii) for each ζ <; f, Ye is a cardinal, J^ is a set, and Eζ is an ultra-filter on Jζ)

(iv) for each ξ < ξ, yζ+1 = yJ

ζζlEζ; and

(v) whenever 0 < ζ ^ ξ, yζ = \Jv<ζΎη+i .

By the ultraproduct of pairs of cardinals (ai9 β{) with ie I, in symbols,Πiei (ocif βi)ID, we mean the pair of cardinals (ILei <*i/A Πiei ft/^)-Similarly (a, βy/D = (α'/D, ^/Z)). The pair of cardinals (/9, β') is saidto be a ξ-limit ultrapower of the pair of cardinals (a, a') if thereexist functions 7, 7', J, £7, with domain ξ + 1 such that α, 7, J", J57, /3satisfy conditions (i) — (v) and a', 7', J, E, βf satisfy conditions (i) —(v). Finally, Sω(I) denotes the set of all nonempty finite subsets of /,i.e., Sω(I) = {s S I\ 0 < φ) < ω}.

To conclude this section, we shall prove a preliminary result con-cerning the products of cardinals.

LEMMA 1.1. Let 1Φθ and let a{ be infinite cardinals with iel.For each iel, let Ji = {jel\aά ^ α j . Then ILesω<u Hiesα* = ΠieΛ ( J i )

Proof. Let A = {a^iel} and, for each a e A, let Ha = {i e I\ a—a}and Sa = {s e Sω(I) \ ΐlies a{ = a}. It is easily seen that

APPLICATIONS OF ULTRAPRODUCTS OF PAIRS 837

Π Π a t = Π «K ( S Λ )

sesω(i) ies a,eΛ

On the other hand

Π«ί ( J i ) = Π ^ Π a*iJi)

Let ae A and je H*. It is clear that Ha C J^ and J y = J{ for eachi e Hω. Thus

/*) _ aκ[Jj)

To prove the lemma, it is sufficient to show that

It is obvious that tc(Jό) ιc(SΛ), and hence

Since α is infinite and Sα C Sω{Jό), we have

The lemma is proved.

2 Cardinalities of ultraproducts We first state a lemma which isan easy consequence of [2, Th. 1.17], and whose proof shall be omittedhere.

LEMMA 2.1. Let IΦO and let ai be cardinals with iel. Thenthere exists an ultrafilter D on Sω(I) such that

Π « ^ Π (iliei sesω(i) \%es

LEMMA 2.2. Let I Φ 0 and let a{ be infinite cardinals with iel.Then there exists an ultrafilter D on SωSω(I) such that

Π (Πα<)= Π (πsesω(i) \iβs / tesωsω{i) \set

Proof. By Lemma 2.1 there exists an ultrafilter D on SωSω(I) suchthat

(1) π ( π « ^ π (πsesωu) \ies / tesωsω(i) \sβt

It is clear that

( 2 ) π (ππA^ πtes

ωsω{i) \set ies // tes

ωs

ωa) \set ie

Since each a{ is infinite and t is finite, we have

(3) π (ππ<o= π intesωsωu) \set iβs / tesωsω(i) \ie\jt

Let Js = {t 6 SωSω(I) \\Jt = s}. By the general associative law for cardi-nal products,

(4) Π Π aA= Π Π Π«,tesωsω(i) Vΐeuί / sesω(i) tejs ies

Since each s is finite, J s is finite. Again, using the fact that each a{

is infinite, we have

(5) Π Π Π « < = Π Π*<s€£ωU) ί€J s i6s sβSωd) ΐ€s

Putting (l)-(5) together we obtain the conclusion of the lemma.

LEMMA 2.3. Let a be infinite and let β > 0. Then there existsan ultra filter D on β such that aβ = aβ/D.

Proof. If β is finite, then cleary aβ = α:β/jD for every ultrafilter Don β. Suppose β is infinite. By Lemma 2.1 there exists an ultrafilterE on Sω(/3) such that

ocβ^ Π (ocκis))IE.

Since each s is finite, ακ ( s ) = α, and hence

Since /3 is infinite, there is a one-to-one correspondence from Sω(β) ontojS, from which we can obtain an ultrafilter D on β such that aβ <xβ/i).It is clear that aβ/D ^ aβ, and therefore the conclusion of the lemmaholds.

LEMMA 2.4. Let a be infinite. Then a™ξ is a ξ-limit ultrapowerof a.

Proof. Let τ 0 = oc. For each ξ S ξ, let yζ = a^c, Jζ = , and Eζ

be an ultrafilter on Jζ such that 7 ^ / ^ = 7 ^ = aωc. Evidently, forξ <£, Ύζ+i = -^+1 = uωζ and hence 7^+i = ΎJ

ζ

ζIEζ. Suppose that 0 < ξ ξ.Then 7ζ = α ^ = Σ^<^ ω > ? = Un<£<*βh' = Un<f 7,+i. The proof is complete.

3* Classes of pairs of cardinals* Let M be a class of pairs of infi-nite cardinals.

THEOREM 3.1. Suppose M is closed under ultraproducts. If IΦ 0Sω(I)}SM, then(Πse^mΠ e A,Πse*ω(i)Π;esft) e M.

Proof. We first show that

(1) for each t e SJ3.(I), ( π Π «<, Π Π ft) e M .\seί t€s s€ί i€s /

Since aiy βt are all infinite and ty s are finite, we have

π π α < = .π><>and

Π Π ft - Π ft .set ies ieuί

Since \Jte Sω(I), (1) follows. Now, the conclusion of the theorem followsfrom (1) and Lemma 2.2.

THEOREM 3.2. Suppose M is closed under ultraproducts. Supposefurther that I φ 0, {(aif β{) \ i e 1} £ M, and for every i, j e I, ai < aά

implies ft ^ βj. Then

π ( j ) π^, s 6 Π 7 π,

Proof. By Theorem 3.1 it is sufficient to prove that

(1) (Π ocif Π ft) e M for each s e Sω(I) .\i€s iβs /

L e t se Sω(I). L e t s± — {jes \ ΐ l i e s a i = a3) a n d s2 — {jes\ I L e s f t = ft}Since s is finite, sλ Φ 0 and *a =£ 0. We show that s± Γi s2Φ 0. Assumesλ Π s2 = 0. Let j 6 s2 and fc e s2. Since αA < α y, we have ft ^ ft. Onthe other hand, ft < ft, and this is a contradiction. Let j es1 n s2;then

(Π^,Πft)-feft )6M,\i€s i€s /

and (1) holds. The theorem is proved.

COROLLARY 3.3. Suppose M is closed under ultraproducts. Letac> βζ> ζ < & be such that {(aζ, βζ) \ ζ < ξ) £ M, and whenever η < ζ < ξ,Oίη < otζ and ft < βζ. Then

Proof. By Lemma 1.1 and Theorem 3.2.

COROLLARY 3.4. Suppose M is closed under ultraproducts. Leta0, a19 a2, , β0, βlf β2, , be strictly increasing sequences such that{(«., β%)\neω}S M. Then (ΪL<ωan, IL<ωβn)e M.

Proof. By Corollary 3.3.

THEOREM 3.5. Suppose M is closed under ultrapowers. If {a, β) e M,then (αγ, β*) e M.

Proof. By Lemma 2.3.

THEOREM 3.6. Suppose M is closed under ξ-limit ultrapowers. If{a, β)eM then (a&, β?ξ) e M.

Proof. By Lemma 2.4 and its proof.

The results obtained so far in this section in Corollary 3.3 andTheorems 3.5 and 3.6 can be stated more simply if we assume theGeneralized Continuum Hypothesis. The reason for this is because theoperations of cardinal powers and cardinal products become more trans-parent. For the remainder of this section, we assume the GeneralizedContinuum Hypothesis.

THEOREM 3.7. Suppose M is closed under ultrapowers. Let {a, β) e M.Then the following hold:

( i ) // cf(a) = cf(β), then (a+, β+)eM.

(ii) If cf(a) > cf(β), then (α, β+) e M .

(iii) // cf(a) < c/(/S), then (α+, β)eM.

Proof. It is known that a < ay if and only if cf(a) 7. By theGeneralized Continuum Hypothesis, we see that αc / ( α ) = a+. Hence theconclusions of the theorem follow from Theorem 3.5.

EXAMPLES. Suppose M is closed under ultrapowers. If (ωω, ω)eMthen (ωω+1, ω±) e M. If (ω2, ω) e M then (ω2, ωx) e M. If (ωω, ωλ) e M then(ωβ+1, ωx) e M.

THEOREM 3.8. Suppose M is closed under ξ-limit ultrapowers.Let (α, β) e M. Then the following hold:

( i ) If a < 7 and β < 7, then (7, 7) e M .

(ii) // β < 7 ^ cf{ά), then {a, y)eM .

(iii) If a<ΊS cf(β), then (7, β)eM.

Proof. By the Generalized Continuum Hypothesis, if 7 ^ cf(a) thenα l = a, and if 7 > a, then α£ = 7. The conclusions follow fromTheorem 3.6.

EXAMPLES. Suppose M is closed under £-limit ultrapowers. If(ωl9 ω)eM then (ωω, ωω) e M. If (ωω+1, ωλ) e M then (ωω+1, ωω) e M.

THEOREM 3.9. Suppose M is closed under ultraproducts. Thenthe following hold:

( i ) If cf(a) = cf(β) and for every 7 < <x, δ < β, there exist 7', δ'such that 7 < 7' < a, δ < δ' < β, and (7', δ') e M, then (α+, β+) e M.

(ii) If cf(a) < /3 α icί for every 7 < α, theere exists 7' swc/& ίλαί7 < 7' < α and (7', /3) e Af, then (a+, β) e M.

(iii) If cf(β) < α and for every δ < β there exists δ' such thatδ < δ' < β and (a, δf) e M, then (a, β+) e M.

Proof. By the Generalized Continuum Hypothesis, the cardinalproduct of any c/(α:)-termed sequence of cardinals whose union is a isa+. The conclusions follow from applications of this remark and Corol-lary 3.3.

EXAMPLES. Suppose M is closed under ultraproducts. If{(ωω+n, ωn) Ineω) £ M, then (α>ω+ω+1, ωω+1) e M. If {(ωw+1, ω1)\neω}^ M,then (ωω+1, ωx) e M.

4. Applications to model theory. We shall now give a brief intro-duction to those portions of the theory of models which are pertinentto this section.

By a similarity type, or briefly a type, we mean a function T whosedomain is a cardinal different from 0 and whose range is included in ω.Let T be a type such that Γ(0) = 1 and let δ be the domain for T. Asystem 21 = (A, i2e>e<β is said to be a structure of type T if A Φ 0, and,for each ξ < 3, Rξ is a Γ(|)-ary relation over A.

Let L(T) be the first-order predicate logic with identity symbol = ,an infinite sequence of individual variables ^ vlf v2y , a Γ(f)-placedpredicate symbol Pξ for each ξ < δ, the usual symbols for propositionalconnectives and quantifiers, and no predicate or functional variables orindividual constants. We assume the definitions of formula and sentenceare known, as well as the notion of a sentence of L(T) holding in astructure of type T. A class K of structures of type T is said to be

an elementary class if there exists a set Γ of sentences of L(T) suchthat a structure 21 belongs to K if and only if every sentence of Γholds in SI. A class K of structures of type T is said to be element-arily closed if whenever 2Ie/£ and every sentence of L(T) holding in2ί holds in 33, then 23 e K (or, equivalently, K is a union of elementaryclasses). Notice that every elementary class is elementarily closed.

By the pair of cardinals for a structure <(A, iίe>e<δ we mean thepair (κ(A), /c{R0)). We let M(K) = {(a, β) \ a, β are infinite and thereexists 2ί e K such that (a, β) is the pair of cardinals for 21}. Noticethat if (a, β) e M{K), then a^β.

The following lemmas are easy consequences of known results inthe literature (see [2] and [4]).

LEMMA 4.1. If K is an elementary class, then M(K) is closedunder ultraproducts, ultrapowers, and ξ-limit ultrapowers.

LEMMA 4.2. If K is elementarily closed, then M{K) is closed underultrapowers and ξ-limit ultrapowers.

In view of these two lemmas, we have the following model-theoreticapplications of the results of §3:

(A) If K is an elementary class, then all results 3.1-3.9 of § 3apply to M(K).

(B) If K is an elementarily closed class, then the results 3.5-3.8of §3 apply to M(K).

Lemmas 4.1 and 4.2, and thus the statements (A) and (B), can besomewhat improved. This is done by substituting the notion of anelementary class by the more general notion of a pseudo-elementaryclass (i.e., PC, see [2] and [7]) in Lemma 4.1, and substituting theelementarily closed class by the more general union of pseudo-elementaryclasses in Lemma 4.2. Moreover, for any structure 21, the class K ofall structures which are isomorphic to elementary extensions of 21 hasthe property that M(K) is closed under both ultraproducts and £-limitultrapowers. Therefore both (A) and (B) are valid for such classes K.

We shall now state some earlier theorems formulated in terms ofpairs of cardinals which will give some idea of how our results (A) and(B) stand with respect to what was previously known concerning M(K).These earlier results differ from ours in that they depend on d, thedomain of the similarity type T.

(C) (Lowenheim-Skolem-Tarski) Let K be an elementarily closedclass. Let (a, β) e M(K) and let 7 be an infinite cardinal such thatδ^j. Then (7, 7) e M(K). Furthermore, if β^y^a, then (7, β) e M(K).

(D) (Vaught [8]) Let K be an elementarily closed class. Let 3 ^La)

and let a Φ β. If (a, β e M(K) then (ω19 ω0) e M(K).

The following is a corollary of (A) and (C).

(E) Assume the Generalized Continuum Hypothesis. Let K be anelementary class. If 8 ^ a, cf(a) < β, and for every 7 < a there exists7' such that 7 < Y < a and (7', β) e M{K), then (a, β) e M(K).

Results similar to (E), but depending on (A) alone, follow fromTheorems 3.7-3.9.

The following is a corollary of (B) and (C).

(F) Assume the Gereralized Continuum Hypothesis. Let K be anelementarily closed class, and let (a, β) e M(K). If β ^ 7' ^ 7 ^ a andδ ^ 7, then (7, 7') e M(K).

Proof. If β = 7', then (7, 7') e M(K) by (C). On the other hand,if β < 7', then it follows from the Generalized Continuum Hypothesisthat βt = 7'. Therefore by Theorem 3.6 and (B), (α#, 7') e M(K). Since7' ^ 7 ^ α&\ we conclude from (C) that (7, 7') e M(K).

5 Some negative results and open problems* In §4 we presentedsome positive results on M(K) when K is an elementary class. In thissection we shall give some negative results stating that certain otherplausible conjectures about M(K) where K is elementary have counter-examples. We conclude this section by stating some natural openproblems.

One can easily construct an example of an elementary class K suchthat {a, β) e M(K) if and only if ω ^ β = a.

We shall now give an example of an elementary class K such that(α, β) e M(K) if and only if ω ^ β ^ a ^ 2β. (This example is due toR. M. Robinson and was privately communicated to R. L. Vaught, fromwhom the authors learned of it.)

Let 8 = 2, T(0) = 1, and Γ(l) = 2. Let K be the elementary classcharacterized by the sentence

Vx,y[xΦV-*3 z{P0(z) A η {Px{x, z)« > Px{y, z))}] .

Thus, if <A, Ro, #!> e K, the mapping f{a) = {b \ R0(b) and Rλ{a, b)} is aone-to-one mapping of A into the set of all subsets of Ro. From thiswe see that M(K) is the desired class of pairs of cardinals.

From this example we easily see that the following hold. Letβ0 = β and, for each n < ω, let βn+1 = 2βn. Then for each n thereexists an elementary class K such that (a, β) e M{K) if and only ifo) ^ β ^ a ^ βn.

We may also easily give an example, when ω ^ β ^ a, of a typeT (with δ = a) and elementary class K such that (α', /3;) e M(K) if and

only \t a^a',β<L β', and β' S a'.For the last collection of examples we need the following restatement

of a lemma in [4].A cardinal β is said to be nonmeasurable if every countably complete

ultrafilter on β is principal. A structure 31 = <A, iJf>e<« of type T issaid to be complete (see [5]) if, for every finitary relation R on A, wehave Re{Rξ \ξ < δ}. Notice that if 21 if complete, then δ ^ 2κU).

LEMMA 5.1. Suppose SI is complete, K is the smallest elementaryclass containing 51, a = /c(A) ω 5g β = /c(R0). Then the following hold:

( i ) Suppose a is nonmeasurable and (a', βf) e M(K). Then eitherα' = a and β' = β, or a' ^ aω and βr ^ βω.

(ii) Suppose β is nonmeasurable and (af, β') e M(K). Then eitherβ' = /3, or a' ^ aω and βf ^ /3ω.

THEOREM 5.2. Suppose ω ^ β ^ a. Then the following hold:( i ) Suppose a is nonmeasurable, and β < βω. Then there exist

a type T and an elementary class K such that (af', β) e M(K) if andonly if af — a.

(ii) Suppose β is nonmeasurable and a < aω. Then there existsa type T and an elementary class K such that {a, β') e M(K) if andonly if β' = β.

(iii) Suppose a is nonmeasurable, a < aω, and β < βω. Then thereexist a type T and an elementary class K such that (a, βr) e M(K) ifand only if βr = β, and (a', β) e M(K) if and only if a' = a.

Proof. By Lemma 5.1.

Open problems. Let K be an arbitrary elementary class and δ ^ ω.I. Does (ωξ+ω+ζ, ωξ) e M{K) and ω ^ β ^ a imply {a, β) e M(K)Ί

II. Does (ωξ+V9 ωζ) e M(K) imply (ωζ+η, ωζ) e M(K)ΊIII. Does (a, β) e M(K) imply (2*, 2P) e M(K)ΊIV. Does \{ωn, ωQ)\n<ω}Q M(K) imply (ωω, ωQ)e M(K)1Γ. Does (ωω, ω0) e M(K) imply (ωω+1, ω0) e M(K)Ί

Does (ωω+1, ωλ) e M(K) imply (ωω+2, ω,)e M(K)ΊIΓ. Does (ω l f ω0) e M(K) imply (ω2, ωλ) e M(K)t

Does (α>3, ω2) e M(K) imply (α>2, ωλ) e M(K)1

REFERENCES

1. C. C. Chang and H. J. Keisler, Pairs of cardinals for models of a given theory, Abstract61T-134, Notices of AMS, 8 (1961), 275.2. T. Frayne, Anne C. Morel and D. Scott, Reduced direct products, to appear in Fund.Math.3. H. J. Keisler, Ultraproducts and elementary classes, Indag. Math., 2 3 (1961), 477-495.


4. H, J. Keisler, Limit ultrapowers, submitted for publication in the Trans, of AMS.5O M Rabin, Arithmetical extensions with prescribed cardinality, Indag. Math., 2 1 (1959),439-446.6. A. Tarski, Quelques theorernes sur les alephs, Fund. Math., 7 (1925), 1-14.7. A. Tarski, Contributions to the theory of models, I and II, Indag. Math, 16 (1954),572-588.8. R. L. Vaught, A Lδwenheim-Skolem theorem for two cardinals, Abstract 578-58, Noticesof AMS, 8 (1961), 239.

UNIVERSITY OF CALIFORNIA, LOS ANGELES

UNIVERSITY OF CALIFORNIA, BERKELEY, AND

INSTITUTE FOR DEFENSE ANALYSES, PRINCETON.

ON DIRECT SUMS AND PRODUCTS OF MODULES

STEPHEN U. CHASE

A well-known theorem of the theory of abelian groups states thatthe direct product of an infinite number of infinite cyclic groups is notfree ([6], p. 48.) Two generalizations of this result to modules overvarious rings have been presented in earlier papers of the author ([3],[4].) In this note we exhibit a broader generalization which containsthe preceding ones as special cases.

Moreover, it has other applications. For example, it yields an easyproof of a part of a result of Baumslag and Blackburn [2] which givesnecessary conditions under which the direct sum of a sequence of abeliangroups is a direct summand of their direct product. We also use it toprove the following variant of a result of Baer [1]: If a torsion groupT is an epimorphic image of a direct product of a sequence of finitelygenerated abelian groups, then T is the direct sum of a divisible groupand a group of bounded order. Finally, we derive a property of modulesover a Dedekind ring which, for the ring Z of rational integers, reducesto the following recent theorem of Rotman [10] and Nunke [9]: If Ais an abelian group such that Extz(A, T) = 0 for any torsion group Γ,then A is slender.

In this note all rings have identities and all modules are unitary.

1. The main theorem* Our discussion will be based on the fol-lowing technical device.

DEFINITION 1.1. Let ^ be a collection of principal right idealsof a ring R. J?" will be called a filter of principal right ideals if,whenever aR and bR are in ^ " , there exists c e aR Π bR such that cRis in ^ .

We proceed immediately to the principal result of this note.

THEOREM 1.2. Let Aa), A{2), ••• be a sequence of left modules over

a ring R, and set A = ΠΓ=i A(<), An = ΠΓ^+iA(ί). Let C = Σ . Θ Cwhere {Ca} is a family of left R-modules and a traces an index set I.Let f: A—*C be an R-homomorphism, and denote by fω: A—*CΛ thecomposition of f with the projection of C onto CΛ. Finally, let SΓbe a filter of principal right ideals of R. Then there exists aR in^ and an integer n > 0 such that fJaAn) gΞ ΓibRβsf bCa for all but afinite number of a in I.

Proof. Assume that the statement is false. We shall first construct

Received November 29, 1961

847

848 STEPHEN U. CHASE

inductively sequences {xn} £ A, {anR} £ ^ , and {αj £ I such that thefollowing conditions hold:

( \ \ n T? —1 n J?\ 1 ) Q>nJΛ> =. Q'n+l-K'

( i i ) xneanAn.

(iii) fajxn) Φ 0 (mod an+1CaJ .

(iv) fajxk) = 0 tor k<n .

We proceed as follows. Select any aλR in ^ . Then there existsa± e I such that faJjiiAJ ς£ f]bBesf bCai, and hence we may select bR inJ^~ such that /^(αiAj) ς£ 6CΛl. Since ^ ~ is a filter of principal rightideals, there exists a2eaλR[\bR such that a2R&J^, in which case/^(ttiAO <£ α2CΛ l. Hence there exists a?! e α ^ such that fΛl(Xi) Φ 0(mod a2Ca). Then conditions (i)-(iv) above are satisfied for n = 1.

Proceed by induction on w; assume that the sequences {xk} and {αA}have been constructed for k < n and the sequence {α i?} has been con-structed for k ^ n such that conditions (i)-(iv) are satisfied. Now, thereexist β19 , βr e I such that, if a Φ β19 , βr9 then fa(xk) = 0 for allfc<^. We may then select anφβ19 9βr such that fΛn(anAn)(£ Γ\ϋRe& bC*^,for, if we could not do so, then the theorem would be true. Hencethere exists bRej^~ such that fΛn{anAn) qL bCv Since J^ is a filterof principal right ideals, there exists an+1 e anR Π bR such that an+1Ris in &~9 in which case fΛn(anAn) ςt an+1Can. Thus we may select xn e anAn

such that foon{xn) Ξ£ 0 (mod an+1CMn). It is then clear that the sequences{xk} and {ak} for k ^ n and {αfei2} for k ^ n + 1 satisfy conditions (i)-(iv),and hence the construction of all three sequences is complete.

Now write xk = (xk

]), where αsj e A(<). Since xkeakAk, xk

l) = 0 fork > i9 and x{i) — YJζ^xf is a well-defined element of AUΊ. Also, sinceanR a ^»+i^ 2 " , it follows that there exists y™ e A(ί) such that x(ί) =

χ(ί) _|_ . . . + a «) + an+1y{

n

ί}. Therefore, setting a? = ($(ί)) and yn = (i/i0),we see that x = χx+ + xn + αn+1^/w for all n*zl.

It follows immediately from inspection of conditions (iii) and (iv)above that a{ Φ as if i Φ j . Hence there exists n such that fΛn(x) = 0.Writing x — xλ + + xn + αw+1j/n as above, we may then apply fΛn

and use condition (iv) to conclude that fΛJxn) = —a,n+ifan(yn) = 0 (modβn+iCa^), contradicting condition (iii). The proof of the theorem is hencecomplete.

In the following discussion we shall use the symbol | X | to denotethe cardinality of the set X

COROLLARY 1.3 ([3], Theorem 3.1, p. 464). Let J? be a ring, andA = ΓLej R{*\ where Rw ^R as a left .K-module and | J\ ^ ^ 0 . Supposethat A is a pure submodule of C — Σ e Θ Cβ9 where each Cβ is a left R-

ON DIRECT SUMS AND PRODUCTS OF MODULES 849

module and \Cβ\ ^ | J\.x Then R must satisfy the descending chaincondition on principal right ideals.

Proof. Since J is an infinite set, it is easy to see that A ^ Πΐ°=i A{i),where A{i) ^ A, and so without further ado we shall identify A withΐlT=iA{i). L e t / : A ^ C b e the inclusion mapping, and fβ:A-+Cβ bethe composition of / with the projection of C onto Cβ. Finally, setA — TT°° 4 ( ί )

Suppose that the statement is false. Then there exists a strictlydescending infinite chain aλR Ξ2 <*>*& =2 of principal right ideals of R.These ideals obviously constitute a filter of principal right ideals of R,and so we may apply Theorem 1.2 to conclude that there exists n ^ 1and &,-•-,& such that fβ(anAn) £ an+1Cβ for β Φ βu . . . , βr.

Now let C = Cβl 0 0 Cβr; then the projection of C onto Cinduces a ^-homomorphism #: anCjan+1C —• anC'lan+1C

f, where Z is thering of rational integers. Also, the restriction of / to An induces a Z-homomorphism h: anAJan+1An —> anClan+1C. An is a direct summand of A,which is a pure submodule of C, and so An is likewise a pure submoduleof C. Hence h is a monomorphism. We may then apply the conclusionof the preceding paragraph to obtain that the composition gh is a mono-morphism. In particular, \anAJan+1An\ ^ \anC'lan+1C'\ ^ | C'\.

Observe that \Cf\ ^ \J\, since J is infinite and \Cβ\ ^ | J\ for all/3. However, since anR Φ an+1R, anR/an+1R contains at least two elements;therefore | anAJan+1An \ = | anAjan+1A \ 2 | J | > | J\. We have thus reacheda contradiction, and the corollary is proved.

2 Applications to integral domains. Throughout this section Rwill be an integral domain. If C is an i2-module, we shall denote themaximal divisible submodule of C by d(C). In addition, we shall writeRωC — Π uC, where a traces the nonzero elements of R.

Our principal result concerning modules over integral domains isthe following theorem.

THEOREM 2.1. Let {Aw} be a sequence of R-modules, and set A =ΠΓ=i^ ( ί\ A-n = ΠΓ=«+iA(<). Let C = Σ * © C * , where each CΛ is an R-module. Let f: A—> C be an R-homomorphism, and fa: A-^Ca be thecomposition of f with the projection of C onto Ca. Then there existsan integer n^l and aeR, a Φ 0, such that afa,(An) £ RωCcύ for allbut finitely many a.

Proof. Let j ^ ~ be the set of all nonzero principal ideals of R.Since R is an integral domain, it is clear that ^ is a filter of principalideals. The theorem then follows immediately from Theorem 1.2.

1 A is a pure submodule of C if A (Ί aC = aA for all a€R.


COROLLARY 2.2 (see [4].) Same hypotheses and notation as in Theorem2.1, with the exception that now each CΛ is assumed to be torsion-free.Then there exists an integer n ^ 1 such that fa(An) g d(Ca) for all butfinitely many a. In particular, if each CΛ is reduced (i.e., has no di-visible submodules) then fa(An) = 0 for all but finitely many a.

Proof. This follows immediately from Theorem 2.1 and the trivialobservation that, since each CΛ is torsion-free, RωCcύ — d(Ca).

Next we present our proof of the afore-mentioned result of Baumslagand Blackburn concerning direct summands of direct products of abeliangroups ([2], Theorem 1, p. 403.)

THEOREM 2.3. Let {Aw} be a sequence of modules over an integraldomain R, and set A = JlT=1 A

{i), C = ΣΠ=i θ A{i) (then C is, in theusual way, a submodule of A.) If C is a direct summand of A, thenthere exists n ^ 1 and a Φ 0 in R such that aA{ί) £j d{A{i)) for i > n.

Proof. Assume that C is a direct summand of A, and let /: A —* Cbe the projection. Then the composition of / with the projection of Conto A{ί) is an epimorphism f{\ A—*A{i). We then obtain from an easyapplication of Theorem 2.1 that there exists n ^ 1 and a Φ 0 in R suchthat afi(A) S RωA{i). Since each ft is an epimorphism, it follows thataA(ί) g RωA(i) for i > n.

Now let z e RωAH), where i > n. If b Φ 0 is in R, then there existsx e A{i) such that abx — z. Hence, setting y = ax, we have that y e RωA{ί)

and by = z. It then follows that RωA{i) is divisible, and so RωA{i) Cd(A{i)). Therefore aA{i) g RωA{ί) g d(A{ί)) for i > n, completing theproof of the theorem.

We end this section with a proposition which will be useful in theproof of some later results.

PROPOSITION 2.4. Let {Aw} be a sequence of finitely generatedmodules over an integral domain R, and set A = ΐ[T=iA(i). Let C =Σ * θ CΛ, where each Ca is a finitely generated torsion iϋ-module. If/: A —• C is an 12-homomorphism, then there exists ce R such that cf(A) =0 but c Φ 0.

Proof. As before we let ^~ be the filter of all nonzero principalideals of R. Clearly RωCai = 0 for all a, and so we may apply Theorem2.1 to obtain a φ 0 in R and an integer n > 0 such that afa(An) = 0for all but finitely many a, where An = IL°°=«+i AH) and fa: A—>Ca isdefined as before. Say this condition holds for a Φ au « , ^ r ; then,since each CΛ is finitely generated and torsion, there exists af Φ 0 in Rsuch that a'CΛi = 0 for i = 1, , r, in which case aaf(An) — 0. Since

each A{i) is finitely generated and C is a torsion module, there existsα" Φ 0 in R such that a"f{A[i)) = 0 for i g n. Set c = ααV; thenc =£ 0 and, since A = A{1) © 0 Ain) 0 An, it is clear that cf(A) = 0,completing the proof of the proposition.

3 Applications to Abelian groups This section is devoted to adiscussion of the results of Baer, Rotman, and Nunke mentioned in theintroduction.

THEOREM 3.1 (see [1], Lemma 4.1, p. 231). Let {A{i)} be a sequenceof finitely generated modules over a principal ideal domain R, and setA — ΠΓ=iΆ(<)- If C is a torsion B-module which is an epimorphic imageof A, then C is the direct sum of a divisible module and a module ofbounded order.

Proof. For each prime p in R, let Cp be the p-primary componentof C and Cp be a basic submodule of Cp (see [5], p. 98;) i.e., C'p is adirect sum of cyclic modules and is a pure submodule of CP9 andCP\C'P is divisible.2 Set C" = Σ P 0 C£; then, since C = Σ P 0 Cp, C" is apure submodule of C and C/C is divisible. Also, C" is a direct sum ofcyclic modules.

We now apply the fundamental result of Szele ([5], Theorem 32.1,p. 106) to conclude that Cp is an endomorphic image of Cp for eachprime p, from which it follows that C is an endomorphic image of C.Since by hypothesis C is an epimorphic image of A, we then see thatthere exists an epimorphism /: A —> C\ By Proposition 2.4, there existsc Φ 0 in R such that cC — cf(A) — 0; i.e., C has bounded order. SinceC" is a pure submodule of C, we may apply Theorem 7 of [6] (p. 18)to conclude that C is a direct summand of C. Since C/C is divisible,the proof is complete.

For the case in which R is the ring of rational integers, the as-sertion of Theorem 3.1 follows from the work of Nunke [9].

In the remainder of this note, R will be a Dedekind ring which isnot a field. If A and C are iϋ-modules, we shall write Ext (A, C) forExtβ(A, C). The following two lemmas are well-known, but to ourknowledge have not appeared explicitly in the literature.

LEMMA 3.2. Let a Φ 0 be a nonunit in R, and let A and C be R-modules. Assume that aC = 0, and a operates faithfully on A (i.e.,ax = 0 for xeA only if x = 0.) Then Ext (A, C) = 0.

2 The definition and properties of basic submodules used here, as well as the theoremof Szele applied in the following paragraph, are in [5] given only for the special case inwhich R is the ring of rational integers. However, it is well-known that these results canbe trivially extended to modules over an arbitrary principal ideal domain.


Proof. Since a operates faithfully on A, we obtain the exactsequence—

0 > A -^U A > Ala A > 0

where ma is defined by ma{x) — ax. This gives rise to the exact coho-mology sequence—

Ext (A, C) i Ext (A, C) > 0

where m*(%) = au for u in Ext {A, C). But, since aC = 0, we havethat mt — 0, and so it follows from exactness that Ext (A, C) — 0, com-pleting the proof.

LEMMA 3.3. Let a Φ 0 he a nonunit in R, and A, C be R-modules.Assume that a operates faithfully on A. Then the following statementsare equivalent:

(a) a operates faithfully on Ext (A, C).(b) The natural mapping Horn (A, C)—> Horn (A, C/aC) is an epi-

morphism.

Proof. Consider the exact sequence—

0 >Ca >C^^C >C/aC >0

where Ca = {x e C/ax = 0} and ma is defined as in Lemma 3.2. Thissequence may be broken up into the following short exact sequences:

0 >Ca >C-^->aC >0

0 >aC-^->C >ClaC >0

where v is the inclusion mapping and μ differs from ma only by theobvious contraction of the range. Since aCa — 0 and a operates faithfullyon A, we obtain from Lemma 3.2 that Ext {A, Ca) = 0, and so the relevantportions of the resulting cohomology sequences are as follows:

0 > Ext {A, C) - ^ Ext {A, aC) > 0

Horn (A, C) > Horn (A, C/αC) > Ext (A, aC) - ^ Ext (A, C) .

Since ma — vμ, we have that ma* — v*μ*, where ma*: Ext (A, C)—>Ext (A, C) is defined by ma*(u) = au for u in Ext (A, C). Hence (a)holds if and only if ma* is a monomorphism. But this is true if andonly if v* is a monomorphism, since μ* is an isomorphism. But it isclear from the second exact sequence above that v* is a monomorphismif and only if (b) holds. The proof is hence complete.

In the remainder of this section we shall set Π = ΠΓ=i^(ί)> where

THEOREM 3.4. Iêt R be a Dedekίnd ring, and a Φ 0 be a nonunitin R. Set C = Σ?=i Θ R/anR- Let A be a torsion-free R-module satis-fying the following conditions:

(a) Every submodule of A of finite rank is protective.(b) a operates faithfully on Ext (A, C).

Then, if fe Hom (77, A), f(Π) has finite rank.

Proof. Assume that the statement is false for some fe Hom (77, A).Then /(77) contains a submodule Fo of countably infinite rank. LetF = {x e A\anx e FQ for some n}. Then F likewise has countably infiniterank. We may then apply condition (a) and a result of Nunke ([8],Lemma 8.3, p. 239) to obtain that F is protective, and then a resultof Kaplansky ([7], Theorem 2, p. 330) to conclude that F is free. Letxu x2, be a basis of F. Then there exist nonnegative integers vu v2,such that yn = aVnxn is in Fo.

Let zn generate the direct summand of C isomorphic to R\anR, andlet zn be the image of zn under the natural mapping of C onto C = CjaC.Define an ϋ!-homomorphism Θ^F—^C by #i(ίcn) = zn+vn- Observe thatθâF) = 0, and so θx induces a homomorphism θ2: F/aF —* C. Now, itfollows easily from the construction of F that the sequence 0 —> FjaF —>AjaF-ÂjF—^0 is exact, and a operates faithfully on A\F. We maythen apply Lemma 3.2 to conclude that this sequence splits. It is thenclear that θ2 can be extended to a homomorphism θ: A —* C. We empha-size the fact that θ(xn) = zn+v

Since a operates faithfully on Ext {A, C), we may now apply Lemma3.3 to obtain cpeHom(A, C) such that the diagram—

\iC

is commutative. Observe that, since θ(xn) = zn+^n, <p(xn) = zn+Vn (mod aC).That is, the coefficient of zn+Vn in the expansion of φ{xn) is 1 + atn forsome tn e R. Since yn = aVnxn, the coefficient of zn+^n in the expansionof φ(yn) is αVw + av^+1tn.

Set ^ = φf\ then sreHom(77, C), and so we may apply Proposition2.4 to conclude that cg(Π) = 0 for some c Φ 0 in iϋ. Since each yn isin /(77), and zn generates a direct summand of C isomorphic to R/anR,it then follows from the preceding paragraph that c(a^ + aVn+1tn) is inan+VnR for all n, in which case c(l + atn) is in anR for all ^. Let P


be any prime ideal in R containing a; then 1 + atn is a unit modulo Pn

for all n > 0, and so c e Pn for all n. Therefore c = 0, a contradiction.This completes the proof of the theorem.

COROLLARY 3.5. Let R be a Dedekind ring (not a field,) and letA be an R-module with the property that Ext (A, C) = 0 for any torsionmodule C. Then, if fe Horn (Π, A),f(Π) is a protective module offinite rank.

Proof. We may apply a result of Nunke ([8], Theorem 8.4, p. 239)to obtain that A is torsion-free and every submodule of A of finiterank is projective. The corollary then follows immediately from Theorem3.4.

The following special case of Theorm 3.4 was first proved byRotman ([10], Theorem 3, p. 250) under an additional hypothesis whitchwas later removed by Nunke ([9], p. 275.)

COROLLARY 3.6. Let A be an abelian group such that Ext (A, C) —0 for any torsion group C. Then A is slender.3

Proof. We need only show that, for any /eHom(/7, A), f(Π) isslender. By Corollary 3.5, f(Π) is free of finite rank. But it is well-known that a free abelian group is slender (see [5], Theorems 47.3 and47.4, pp. 171-172.) The proof is hence complete.

REFERENCES

1. R. Baer, Die Torsionsuntergruppe Einer Abelschen Gruppe, Math. Annalen, 135(1958), 219-234.2. G. Baumslag and N. Blackburn, Direct summands of unrestricted direct sums ofAbelian groups, Arkiv Der Mathematik, 1O (1959), 403-408.3. S. Chase, Direct products of modules, Trans. Amer. Math. Soc, 97 (1960), 457-473.4. , A remark on direct products of modules, Proc. Amer. Math. Soc, 13 (1962),214-216.5. L. Fuchs, Abelian Groups, Publishing House of the Hungarian Academy of Sciences,1958.6. I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, 1954.7. 1 Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc,72 (1952), 327-340.8. R. Nunke, Modules of extensions over Dedekind rings, 111. Math. J., 3 (1959), 222-241.9. , Slender groups, Bull. Amer. Math. Soc, 67 (1961), 274-275.10. J. Rotman, On a problem of Baer and a problem of Whitehead in Abelian groups,Acta. Math. Sci. Hung., 12 (1961), 245-254.

PRINCETON UNIVERSITY

8 For the definition of a slender Abelian group we refer the reader to [9].

ANNIHILATORS IN THE SECOND CONJUGATEALGEBRA OF A GROUP ALGEBRA

PAUL CIVIN

l Introduction* Let © denote an infinite locally compact abeliangroup, and let L(@) be its group algebra. The second conjugate spaceL**(©) of the group algebra can also be considered as an algebra bythe use of Arens multiplication [1] [2]. Civin and Yood [3, p. 857] haveshown that L**(@) is an algebra which is not commutative and has anonzero radical 9ΐ**. They have also shown [3, p. 856] that if © is notdiscrete, then the algebra L**((S) has a nonzero right annihilator.

The object of the present note is the study of the nature of theleft and right annihilators of the maximal modular left ideals in L**(@).It is shown that such annihilators are either nilpotent two-sided or rightideals, respectively, or else the maximal modular left ideal in questionmust have the form {Fe L**(®)\F(μ) = 0} where μ is some multiplica-tive linear functional on L((S). If © is compact it is seen that all maximalmodular left ideals of the latter form have a nonzero left annihilator anda right annihilator which properly contains the right annihilator of L* *(($).

It should be noted that the choice of the maximal modular left idealsas the subject of investigation is not simply for definiteness. At thepresent stage of available information concerning L**((S), the maximalmodular left ideals are more tractable than the corresponding right ideals.

2 Notation* Throughout the note we shall use the notation in-troduced above as well as other notation introduced by Civin and Yood[3], In particular 9ΐ** will denote the radical of L**(@) and 2) will denotethe closed subspace of L*(@) generated by the multiplicative linear func-tionals on L(©). We shall write 2(7) (3ΐ(/)) for the left (right) annihilatorsin the algebra L**(@) of the subset /of L* *(($). We also use the nota-tion / x ( / τ ) for the linear space annihilator in B*(B) of the linear manifoldI in the Banach space B (the conjugate space £*). Throughout π willbe used for the natural embedding of a Banach space B into its secondconjugate space I?**. It should be recalled [1] that when B is a Banachalgebra, π is an algebra homomorphism, and if B is commutative then[3, p. 855] πB is in the center of B**.

3. Left annihilators* Throughout this section we let Ϊ5R denote amaximal modular left ideal in L**(@) for which S(2K) Φ (0).

LEMMA 3.1. 3JΪ and 8(5Ui) are 2-sided ideals in L**(@) and

Received November 22, 1961. This research was supported by the National ScienceFoundation grant NSF-G-14111.

855

856 PAUL CIVIN

= 9JΪ.

Proof. Since 2JΪ is a left ideal, 8(50Ϊ) is a 2-sided ideal. Thus 3ft8(50ϊ)is a 2-sided ideal containing 50Ϊ. However, the algebra L**(@) contains[3, p. 855] a right identity Ey so £(50Ϊ) Φ (0) implies 3ϊ£(2Jί) is proper, hence318(501) = 50Ϊ, and 9JΪ is a 2-sided ideal.

In the next several lemmas we consider the consequences of theassumption 8(501) qL m.

LEMMA 3.2. If S(2Jί) <£ 2Ji, then S(5Dl) = (L**(©))A, wiίft A = A2.

Proo/. It follows from £(SJί) £ 501 that £**(©) = S(50Ϊ) + 9Ji. Thusthe right identity E satisfies E = A + Λf with A € 2(50i) and Λf e 5K.Left multiplication by F G S(50Ϊ) yields .F = F2? = FA, so in particularA = A2 and 8(50ϊ)c(L**(©))A. The reverse set inequality is immediatesince £(50Ϊ) is a left ideal.

We adopt as fixed notation E = A + M, with A G S ( T O ) and M G 9 K ,

throughout the section in which we are discussing 8(2Jί) φ 501.

LEMMA 3.3. For αiί Fe L**(®), A F = AFA.

Proof. As above £7 = A + ikf. Left multiplication by AF gives= AFA since A e S(3)ΐ) and FMe 501.

LEMMA 3.4. // £(9Ji) ς£ 501, ίfee^ A80K) is ίfee seί o/ complex multi-ples of A.

Proof. Let L Φ 0 be an element of AS(9Ji). Then by Lemma 3.3,L = AL = ALA. Since L Φ 0 and A 6 8(501), it follows that LA 0 501 andL $ 50ί Consequently (L**(@))LA is a left ideal not contained in "SI. HenceL**(@) = 3Jί + (L**(©))LA, and E = N + CLA, with iNΓe 9JΪ. Left multi-plication by A, and appropriate use of the right identity yields A =ACLA = ACFLA = AC(A + M)LA = ACALA = (ACA)(ALA). Thus thenormed algebra Aδ(3Ji) has A as an identity and each nonzero elementhas a left inverse. This implies that Aδ(3ϊί) is a complex normed divi-sion algebra and the lemma then follows from the Gelfand-Mazur theorem.

LEMMA 3.5. If 8(9Ji) ς£ 50i, then there exist a multiplicative linearfunctional ψ on L**(@) such that 3Ji = {Fe L**(®)\φ(F) = 0}.

Proof. In view of Lemma 3.4 and the fact that AS(9Ji) is a rightideal, we may define the complex number φ(F) for Fe L**(@) by AF=φ(F)A. Clearly φ is additive and by the use of Lemma 3.3 we see that

ANNIHILATORS IN THE SECOND CONJUGATE 857

φ(FG)A = AFG = AFAG = φ(F)φ(G)A, so ψ is multiplicative. It followsfrom Lemmas 3.1 and 3.2 that φ(F) = 0 if and only if FeVJl.

THEOREM 3.6. Let 2JΪ be a maximal modular left ideal in L**(@)with 8(50Ϊ) Φ (0). Then m is a 2-sided ideal and either (S(3Jϊ))2 =(^(SJl))2 = (0) or there exist a multiplicative linear functional μ on L(©)such that 2Jί = {Fe L**(®)\F(μ) = 0}. In the latter case S(50i) is a one-dimensional 2-sided ideal in L**(©) and 5R(2Jί) = 3ΐ(L**(©))0S(3K).

Proof. If (0) Φ 8(2») c 501, then 30Ϊ = 9ΐS(23ϊ) 3 St(2K), so (8(501))" =(3ΐ(2Jί))2 = (0). If 8(3Jί)£9Jί, let <pe £***(©) be the multiplicative linearfunction on L**(@) whose existence is guaranteed by Lemma 3.5. Sinceπ is a homomorphism, the functional μ = φoπ is a multiplicative linearfunctional on L(©). The null space of μ is then either L(@) or a modularideal 9JΪ* in L(@). If the first possibility prevails, ττL(©) c 3Jί, and thus0 = A{πx) = (7rα)A for all a? e L(@). The w*-density of τrL(©) togetherwith the w*-continuity of left multiplication [2] in L**(©) implies thatFA = 0 for all Fe L**(@). This contradicts A = A2 Φ 0. We thus con-clude that there is a maximal modular ideal SUί* in L(@) such thatπ(jΰl^(z(ίSl. Now [3, p. 865] the w*-closure of πSJΪ* is a maximal modularleft ideal 3Jί0 in L**(©). Let F G 2 « 0 , then F = w*-\imπxa,xΛe'>ΰl*.Thus 0 = A(πxa) = (τrajΛ)A for all α, so FA = 0, i.e. A e R(Wl0). There-fore by Lemma 3.5, <p(F)φ(A) = φ(FA) = 0. However, since A $ 2JΪ,φ(A)=£0 and consequently φ(F) = 0, so .Pe9Ji. Therefore 3Jίoc5Dΐ andSDΐ = 2fl0. In particular A e 3ft(2K). Also if F G 9JΪ, ί 7 = w * - lim πxa,xΛ 6 SJi and thus F(μ) = lim πxa(μ) = lim /£(»,») = lim φ(πxa) = 0. Sincethe set of jFeL**(@) such that -FXμ) = 0 is a maximal modular idealcontaining 2W, we see that 501 has the appropriate form.

It now follows from Lemma 3.4 that £**(©) = 3JΪ0A8(2Jί) with thesecond summand one-dimensional. Since A e 3ΐ(9Jί), it then follows fromLemmas 3.2 and 3.3 that 8(501) = A2(W) and so is a one-dimensional 2-sided ideal. Since Ae9ί(3K) we have SR(L**(©))©8(aJl)c9ί(3Dΐ). Alsoif .Fe3i(3JΪ), then F = Aζ + a A, with ik^e 3Ji and α complex. SinceAe3l(50*)ΓlS(50i), it is immediate that M^fRiL*^®)) which completesthe proof.

4* Right annihilators* Again we let 9JΪ denote a maximal modularleft ideal. If © is not discrete [3, p. 856] then (0) Φ 9ΐ(L**(©)) c gft(2»).On the other hand we saw in Theorem 3.6 that if (0) Φ (8(9K))2 then3t(2R) = 3ΐ(L**(@))φS(aJi). Our object in this section is to investigaterelationships between 9Ji and 9ϊ(3Jί) with no hypothesis on 8(501). Asindicated in the introduction, we use Sft** for the radical of L**(@).

4.1 LEMMA. Either 3t(5Oί) c 3R** or ίΛere eajisίs an F e 3t(50ϊ)

8 5 8 PAUL CIVIN

is not left quasi-regular.

Proof. Suppose that the right ideal 5R(9Ji) is left quasi-regular. LetFe3ΐ(3Ji). Since FD is left quasi-regular for all Z)eL**(@), we see [5,p. 17] that (L**(©))JFϊis a left quasi-regular left ideal, and so is a quasi-regular left ideal and is included in 3ΐ**. Thus EFeW*. However,

9t**, so

4.2 LEMMA. // 3ΐ(aJi) ςz! 3ΐ**, there exists an AG3ΐ(3Jί) such that

Proof. By Lemma 4.1 there is an Fe 3R(3Ji) which is not left quasi-regular. The left ideal {BF — i?|i?eL**((S)} is then a proper modularleft ideal, so is contained in a maximal modular left ideal 9 . It followsfrom BF=0 for JBe9Ji that 9ίί = Sΰl. Consequently F'-FeWl andtherefore F2 = F* = F\ Thus A = F2e Sft(gM), and A =£ 0 since other-wise F would be left quasi-regular.

We fix the notation in the remainder of this section so that A hasthe properties asserted in the lemma.

4.3 LEMMA. // 91(30?) £ 9Ϊ**, then( i ) E= N+ A, Ne2Rf

(ii) £**(©) = 50Ϊ©(L**(@))A, and(iii) (L**(®))A is a minimal left ideal of L**(®).

Proof. Since A has the properties asserted in Lemma 4.2, A 0 Λίand therefore L**(@) - 2Jl©(L**(@))A with the sum clearly a directsum. Let E = ΛΓ+ -BA with NeWl. Right multiplication by A yields£Ά = BA. Thus BA - A = EA - A e SR(L**(©)) c 2Jί. Another rightmultiplication by A yields J?A = A, so E — N + A.

Suppose that (0) Φ % is a left ideal in (L**(®))A. Then L**(@) =Afφ3f. Let jBeL**(@). Then £A = Mx + I, with ik^ 6 3Ji and ^eQf.Right multiplication by A shows that BA — Iλ so (L**(®))A is a minimalleft ideal.

LEMMA 4.4. 1/ 3ΐ(3Dΐ)ς£ 31**, then there exists a <?eL***(@) suchthat for each XeL**(@), (AX)2 =

Proo/. Since (L**(®))A is a minimal left ideal, A(L**(®))A is adivision algebra and so by the Gelfand-Mazur theorem consists of thescalar multiples of A. For XeL**(®), define φ(X) by AXA = φ{X)A.As defined φ is clearly linear. Moreover, \φ(X)\\\A\\ = \\φ(X)A\\ =| |AXA||g| |A| | 2 | |X| |, so \φ(X)\ £ \\A\\ \\X\\ and φ e L***(®). Theremaining assertion is now immediate.

4.5 LEMMA. Let <m* = {xe L(@) |πx e 2JΪ}. If 3ΐ(9K) φ 91**, then 2fl*is a maximal modular ideal of L(@).

Proof. Note first that 2(A) is a left ideal containing 3JΪ, and A $ 2(A).Therefore 2(A) — Sΰl. It is an immediate consequence of the definitionof φ given in the proof of Lemma 4.4 that φ is multiplicative on thecenter of L**(@). Since ττL(@) is central it follows that φoπ is a multipli-cative linear functional on L(@), and so has a null space which is eitherall of L(@) or is a maximal modular ideal of L(@). Now (πx)A = A(πx)A=φ(πx)A, so φ(πx) = 0 if and only if πx e 2(A) = 2JΪ or if and only ifx e 2Ji*. If 2ft* were all of L(@), then the w*-continuity of left multiplica-tion in L**(©) together with the w*-density of πL(®) would imply thatA2 = 0 which is not the case. Thus 9JΪ* is a maximal modular ideal of!/(©) as asserted.

We will use in the sequel two lemmas which are valid in the algebra5** of the second conjugate space of a Banach algebra B. The nota-tion is that of [3].

4.6 LEMMA. A w*-closed subspace $ of U** is a left (right) idealof 5** if and only if </, x> e 3 T for all f e $τ and x e B ([F,f] e 8 T

for all / e $ τ and Fe B**).

Proof. The argument will be given only for left ideals. Suppose8 is a left ideal and let / e $ τ and x e B. Then for any Fe g, (πx)Fe 3f,so 0 = (πx)F(f) = F(ζf, α?». Consequently </, # > e $ τ . Suppose nextthat fe $ τ and a? e JB implies </, x> e $ τ . Then for .F7 e $ and ίDGΰ,0 = F(ζf, a?» = (πx)F(f), so (TΓCC)FG T J - = $. The ^;*-density of τr#in S** together with the w*-continuity of left multiplication and the^*-closure of $ give HFe % for all He$ for all HeB**, so % is aleft ideal in 5**.

4.7 LEMMA. If $ is a left ideal in JS**, then so is $ τ l .

Proof. The subspace $ l τ is w*-closed. If / e $ τ and xeB, thenfor any Fe%, (πx)Fe$ so 0 = (πx)F(f) = F«J, xy) and </, xye$Ύ.Since $5T = $ τ l τ , Lemma 4.6 yields the desired conclusion.

4.8 LEMMA. If Wl is a maximal modular left ideal of L**((S) with!3t**, then 2JΪ is w*-closed.

Proof. In view of Lemma 4.7, if 3JΪ were not w*-closed, L**(©) =and then (0) = 3ftTJ-τ = 3Jiτ. If A has the same meaning as in the

earlier lemmas, A2 Φ 0, so there is an /oeL*(@) such that [A.,/o] ^ 0.However, since Ae5R(9Jί), [A,/0]6 5Dΐτ. Thus 50i is w*-closed.

860 PAUL CIVIN

4.9 THEOREM. Let 9Ji be a maximal modular left ideal in L**(©).Then either ffiiW))2 = 0 or there exists a multiplicative linear functionalμ on L(@) such that 20Ϊ = {F e L**(β)\F(μ) = 0}.

Proof. If 3ΐ(3Dΐ)c9ϊ**, then 9ΐ(9Dΐ)c2ϊϊ and (9ΐ(3Dΐ))2 = 0. Supposethat $R(3Jϊ) ςί 2JΪ. Let μ be the multiplicative linear functional on L(©)corresponding to the maximal modular ideal 2R* of Lemma 4.5. ByLemma 4.2 there is an A e 3t(3W), 0 =£ A = A2. Let $ be the closed spanof {[A,/]|/eL*(®)}. Since A e S R ( 5 I K ) , ί ϊ c ^ ^ s o ^ i ^ a R . Also if ^ e ^and α? e L(©), then # = lim[A, gn] and <#, #> = lim<[A, gn], x) = lim[A,<Λ, »>], so <g, x)e Jϊ. Thus by Lemma 4.6, 5S1 is a left ideal in L**(©).Since S 1 => 2ft, either K 1 = 2ft or = L**(@). The latter is impossible sinceA2 Φ 0, and thus S 1 = 2JΪ. Now if xeWl* then TΓO; G 2JI, so xe®τ. Thus$ τ c SDΐ and S c S T i c 9Jϊi. However since the latter set consists of thescalar multiples of μ, so also must $. Thus 2Ji has the indicated form.

5. Existence* The question of the existence of maximal modularleft ideals in L**(@) with SR(aW) £ 3ΐ** or with £(3Ji) £ 3ΐ** is easilyresolved if © is compact. For © not compact, necessary and sufficientconditions are given for the existence of ideals with the indicated prop-erties, but no conclusion is reached as to whether or not the given con-dition is automatically satisfied.

5.1 THEOREM. Let © be an infinite compact abelian group, and letμ be a multiplicative linear functional on L(@). Let 9Ji = {F e L**(©)|F(μ) = 0}. Then 3ΐ(90ΐ) £ 3ΐ** and 2(Wΐ) <£ 3ΐ**.

Proof. Since © is compact, its character group is discrete. Theregularity of the Banach algebra L(@) then implies that there is ane e L(©) such that μ(e) = 1 and v(e) = 0 for every multiplicative linearfunctional v on L(©) with v Φ μ. The semi-simplicity of L(©) thenimplies e ~ e2 Φ 0. Since πe is an idempotent in L**(©) and thus πe $ 9ΐ**,it suffices to show that πe e 8(501) Ω 9ϊ(9Ji). Also since πe is central itsufficies to show πe e 3ί(3Dΐ). Now for πx e 9Jί, v{xe) = 0 for all multiplica-tive linear functional v on L(@) so xe = 0 and (π#)(7re) = 0. However,3Jί is [3, p. 865] the ^^-closure of {πx \ πx e 9Jί}, so the w*-continuity ofleft multiplication shows that πe e 9ϊ(9Ji) as desired.

5.2 LEMMA. Let μ be a nonzero multiplicative linear functionalon L(@). Then there exists ΰeL**(@) such that D(μ) = 1, while if vis a multiplicative linear functional on L(©) and v Φ μ, then D(μ) = 0.

Proof. We use the notation for multiplicative linear functionals onL(©) corresponding to the interpretation of the functional as a member

of the character group ©. Let M donote the almost periodic mean. Thenfor any multiplicative linear functionals μζ on L(®), μi Φ μ and for anycomplex numbers ai9 i = 1, , n,

M(l - ta.μ-'βΛl =1

where the norm is that of L*(@). Thus the distance from μ to the spanof the other multiplicative functions is at least one. The desired func-tional De L**(@) then exists as a consequence of the Hahn-Banach theo-rem. The author is indebted to a referee for the suggestion of the aboveproof for Lemma 5.2.

5.3 THEOREM. Let 2) be the closed subspace of L*(®) generated bythe multiplicative linear functionals on L(@). Let Q be the closed spanof {[F,f]\Feψ and /eL*(©)}.

( i ) A necessary and sufficient condition that there exist a maximalmodular left ideal TO in L**(@) with 8(TO) <£ 31** is that 2) qL Q.

(ii) A necessary and sufficient condition that there exist a maximalmodular left ideal TO in L**(@) with 3t(2W) £ 3ΐ** is that thereexist Btβi}1 such [B,f] e 2) for all f e £,*(©).

Proof. Suppose first that there exists a maximal modular left idealTO in L**(@) with «(TO)<Z$R**. Then by Theorem 3.6 there exists amultiplicative linear functional μ on L(®) such that Wl = {Fe L**(®)\F(μ) =0}. By Lemma 3.2, 8(TO) = (L**(®))A and A2 = A Φ 0, so A 0 TO. Itfollows that A(μ) = 1. Suppose that 2 ) c 3 , so that μ e g . Thus

iM = Km S [<?,,„/..*]n i=l

with G,,4 e V)1. Now 2)-1- c SDΪ and A e 80K) so A e S^-1-). Thus

1 = A(μ) = l im"ΣAG..^/. . , ) = 0 .

Consequently 2) ζί 3Suppose that 2) ζί ,3 Then there exist some multiplicative linear

functional μ on L(@) such that μφ $. Thus there exists J e L**(®) suchthat J e 3 1 and J(/^) = 1. Let jDeL**(@) have the property assertedin Lemma 5.2. Let TO = {F e L**(@) | F(μ) = 0}. Clearly TO is a maximalmodular left ideal of L**(®). Let i ϊ = JD. Then H(β) = J(μ)D(μ) = l,so if 0 2H and therefore Jϊ 0 91**. Let P e TO, and let / e L*(®). ThenHP(f) = JDP(f) = J([DP,f]). Now if v is any multiplicative linearfunctional on L((8), (DP)(v) = i)(v)P(i;) = 0, since Lemma 5.2 J9(v) = 0if v ψ μf while P(v) = 0 if v = μ since P e l . Thus DP e 2)\ and thus[DP,f] e 3. However J e 3 \ so i ίP(/) = 0. Since / was arbitrary in

862 PAUL CIVIN

L*(@) and P arbitrary in 2R, we see that H e 8(2K) and Hφ #** whichcompletes the proof of the first half of the theorem.

Next, we suppose that there exist a maximal modular left ideal Sΰlin L**(@) with ίR(W)ςt 3ΐ**. By Theorem 4.9, there is a multiplicativelinear functional μ on L(®) such that Tt = { F G L * ^ © ) ! ^ / * ) = 0}. Alsoby Lemma 4.2, there exists A e ?H(W) such that A — A2 Φ 0. In particular.A 0 9Ji, so A 0 2H as 2) c 2JΪ. Let / e L*(@). Then A e 3t(2») soA G S R ( ^ ) . Thus for any Te?μ, 0=ΓA(/)=Γ([A,/]), and [A,/] G?)T J- =

2). Thus A has the required properties.Finally, we suppose that there exist B $ 2)1 such that [B,f] e2) for

each / G L*((?). Since S 0 2)1, there exist a multiplicative linear func-tional μ such that B(μ) Φ 0. Let Wl = {F e L**(®)\F(μ) = 0}, so that9Ji is a maximal modular left ideal in L**(@). By Lemma 5.2, thereexist A G L**(©) such that A(μ) = 1 and A{v) = 0 if ^ is a multiplicativelinear functional on L(@) different from μ. Now AB(/J*) = A(μ)B(μ) Φ 0,so AS 0 $R**. Let P G 9DΪ, then for / G L*(@), [5, /] e 2), so

where each ^ w i is a multiplicative linear functional on 8((S) and eachcn>ί is a complex number. We choose the notation so that μn>1 = μ. Henceby the stated properties of A and the fact that [A, v\ — A(v)v for anymultiplicative linear functional μ on L(®) we see that

[AB,f] = [A, [B,f]] - linΓj?cnΛA(μ%ti)μnΛ = \imcnΛμ .

Thus PAB(f) = P([AB,f]) = 0, and since / was arbitrary in L*(®) andP arbitrary in 3Jί we have ASG3Ϊ(9K). This completes the proof ofTheorem 5.3.

REFERENCES

1. Richard Arens, Operations induced in function classes, Monat. fiir Math., 55 (1951),1-19.2. , The adjoint of a bilinear operation, Proc. Amer. Math. Soc, 2 (1951), 839-848.3. P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra,Pacific J. Math., 11 (1961), 847-870.4. L. H. Loomis, An introduction to harmonic analysis, D. Van Nostrand Co., New York1953.5. C. E. Rickart, General theory of Banach algebras, Van Nostrand, New York 1960.

T H E UNIVERSITY OF COPENHAGEN

T H E UNIVERSITY OF OREGON

POLYNOMIAL INTERPOLATION IN POINTS

EQUIDISTRIBUTED ON THE UNIT CIRCLE

J. H. CURTISS

U Introduction, Let distinct points Sn = {znl, zn2, , znn} be givenon the unit circle | z | = 1 in the complex 2-plane, let a function / alsobe given on | z | = 1, and let Ln — Ln(f; z) denote the polynomial of degreeat most n — 1 found by interpolation to / at the points Sn. Consideran infinite sequence of such point sets, Slf S2f , Sn, , and the cor-responding sequence Ll9 L2, , Ln, . If the union of the sets Sn iseverywhere dense on \z\ = 1, does lim^ooLn(f; z) exist for | z | < 1, andif so, what is it?

Walsh [14, pp. 178-180] proved that if the points Sn are equallyspaced for each n, and if / is Riemann integrable, then

(1.1)2πι)\t\=i t — z

uniformly on any closed point set on the region | z \ < 1. The presentauthor [1] [2] generalized Walsh's result to the case of interpolation ona more or less arbitrary Jordan curve. The problem for equally spacedinterpolation points has a pedigree of some length which is described inWalsh's book [14] and in a recent survey given by the author [3].

When the points Sn are not equally spaced, very little is knownabout the behavior of Ln unless / is analytic on | z | ^ 1. For the analyticcase Fejer [4] proved that if the points Sn are equidistributed on anarbitrary Jordan curve C in a sense to be described below in §2 andif / is analytic on the closed region D bounded by C then Ln-^f uni-formly on D. No result of this sort involving equidistribution is atpresent known for nonanalytic functions / even when C is the unitcircle.1 It is the purpose of this paper to try to shed some light onthe situation for nonequally spaced points by means of a probabilistictreatment. We shall let the points of the sequence Si, Sa, be randomvariables defined on a probability space with a structure such that almostcertainly a sample sequence is equidistributed. (We use the word"equidistributed" here in connection with sample sequences rather thanthe more usual words "uniformly distributed" to avoid confusion withthe concept of a uniform distribution in the probability sense.) The

Received September 28, 1961. This research was supported by the United States AirForce through the Air Force Office of Scientific Research of the Air Research and Develop-ment Command, under Contract No. AF 49(638)-862.

1 Zygmund [16, vol. II, pp. 3-4] points out that a similar gap exists in the theory oftrigonometric interpolation.

863

864 J. H. CURTISS

mean value of Ln formed in the random points Sn is calculated in §2.The result is consistent with (1.1). But in § 4, in discussing a particularclass of equidistributed sample sequences, we shall show how only aslight modification of equal spacing upsets Walsh's deterministic result.

2Φ A stochastic treatments In this section we shall first use astochastic model which is appropriate to the case in which for each n > 1the first n — 1 points of Sn are the points of Sw_i—or in other words,the first subscripts on the points znk are superfluous.2 A slight extensiongiven in the next section will provide the structure for the situationdescribed in the first paragraph of the Introduction, in which Sn mayconsist entirely of new points not used in $„_!.

Let θlf θ2, , be an infinite sequence of mutually independent randomvariables each with the uniform (or "rectangular") marginal probabilitydistribution on the closed interval [0, 2π], Let zk = eίθfc, k = 1, 2, •••.Let the function / be given everywhere on | z | = 1, and let

(2.1) Ln(f; z) = Ln(f; zl^z*,—, zn) = £ " f i) 7 f i , , >(z - zk)ω'n(zk)

where ωn(z) = (z — zλ)(z — z2) (z — zn). For each sample sequence ofthe stochastic process zl9 z2j , for which the values of z19 z2, * ,zn

are distinct, the formula (2.1) gives the unique Lagrange polynomialof degree at most n — 1 found by interpolation to / in this value. Thelocus of those points in the ^-dimensional interval 0 ^ θj ^ 2π, j = 1, , n,for which (2.1) is formally undefined is the union of hyper planes

U {θj, θk I θj = θk mod 2π} .

The probability measure attached to each such hyperplane is zero, so itfollows that (2.1) defines a Lagrange polynomial with probability one.

By the Glivenko-Cantelli Theorem [9, pp. 20-21], given any samplesequence of the process θl9 θ2, , if Nn(θ) denotes the number of valuesof the first n terms falling into [0, θ], then with probability one

(2.2) J m JLn 2π

uniformly in θ. The condition (2.2) is the classical definition of equi-distribution or deterministic uniform distribution [15], [11, vol. 1, pp.

2 This is the only model considered in [4] where Theorem 1 below is announced with-out proof. It might be noted that Carl Runge, who in 1904 published a proof of the result(1.1) for functions analytic on | z | 5Ξ 1, considered only a sequence of sets of equally spacedpoints in which each set contained all the previous sets. That is, he interpolated in thenth, 2nth, 4nth, etc., roots of unity. See [12, pp. 136-137].

POLYNOMIAL INTERPOLATION IN POINTS EQUIDISTRIBUTED 865

70 ff.], for real numbers. In the work referred to in the Introduction,Fejer translated it to points on an arbitrary Jordan curve C by para-metrizing C through the schlicht analytic function z = φ(w) which givesa conf ormal map of | w | > 1 onto the exterior of C so that the pointsat infinity correspond. The function can be extended in a continuousand one-to-one manner (Osgood-Taylor-Caratheodory Thoerem) onto | w \ =

I I —

The theorem about to be stated and proved deals with the meanvalue of Ln over the marginal distribution of the vector random variable(#i, #2, *',θn) Given any finite subsequence consisting of k membersof the random sequence θ19 #2, , and a function g from the intervalIk = [0, 2π] x [0,2π) x x [0, 2π] (k factors) to the complex planeintegrable in the sense of Lebesgue on Ik9 we shall use the symbol Ekgto denote the mean value

( 1 \kC2π C2π

— ) g(alf a2, , ak)dax dak .2 τ τ / J o Jo

THEOREM 1. If f is continuous on \ z \ — 1 and possesses there an(n — 2)th order derivative satisfying a Lipschitz condition with exponentone, then for all z

(2.3) EnLn(f; z) = ao + aλz + a2z2 + + a^z^1 , n ^ 2 ,

where Σo o>kZk is the Taylor expansion of the analytic function

(2.4) F(z) = - M β$-dt, \z\<l.2π% J ι*ι=i ί — z

Thus iff is infinitely differentiate on \z\ = 1, lim^oo EnLn(f; z) existsand equals F(z) for | z \ 1. If f is analytic on \ z | ^ 1, then thislimit exists uniformly and equals f(z) on each closed disk with centerat the origin which does not contain a singularity of f.

The statements in the theorem following (2.4) are standard resultsin function theory relating to the Taylor expansion of F and to thepossibility of extending F continuously onto | z \ = 1. See for example[14, pp. 141 ff.].

The equation (2.3) is trivially true for n = 1 if / is merely integrableon I z I = 1, because then L^f; zt) = f(z±) and

2πτ Juι=i; *d [f(e)dθi ih \ ψdt

2π Jo 2πτ Juι=i t

The derivatives of / referred to in the theorem may be taken withrespect to arc length (here identical with θ in the parametrization z =

866 J. H. CURTISS

eiθ of the circle | z | = 1), or alternatively they may be taken in a chordalsense:

/'OO = li

and so on for higher derivatives. To fix ideas we shall use the latterinterpretation.

The Lipschitz condition referred to in the theorem means that forsome I > 0,

all \zλ\ = I s3 | = 1. It implies of course that f{k) with 0 ^ k < n — 2,satisfies a similar condition because f{k) has a continuous derivative on| s | = l.

An alternative expression for the right side of (2.3) is given by

(2 5) Ψazk- X

(2.5) Σ akz - _ _

which follows with no hypotheses on / other than integrability fromthe validity of

2πi Jι*ι=i ί VI - ί/jg/ V 2τrί Juι=i

and from the uniqueness of Taylor series. Our proof will establish theequivalence of the left side of (2.3) with the right side of (2.5). Weneed some preliminary results before passing to the main proof.

LEMMA 1.

( 1, m ^ 0, n ^ 0

ΊΓ\ ^~Λ —e dθ = \-l,m<0,n<0

2π Jo 1 — e%θ

{ 0, otherwise,where n and m are integers.

The integrand consists of the sum of a finite number of positiveand negative integral powers of e~ίθ. With m ^ 0 and n ^ 0, the coef-ficient of the zeroth power is one; with m < 0 , n < 0 it is —1; andotherwise there is no zeroth power in the sum at all.

LEMMA 2. Let g{θ) have the period 2π and be such that

rrJo Jo

g(a) - g{β) dadβ

exists. Then

2π ) Jo Jo e%" - e

— [2

2π Jo

— [2πe-i{m+n+1)θg(θ)dθ , n ^ 0 , m ^ 02π Jo

- J _ [**e-iim+n+1)$g(θ)dθ , n < 0, m < 02π Jo

0 , otherwise.

For the proof, we make the change of variables u = β — a, a — a,and using periodicity arrive at

J = (JLY [2π [2π e-^^e-^igja) - g(a + u)) d(χdu

\2πJ Jo Jo 1 - e

ίu

Fubini's theorem, applicable because of the integrability hypothesis,allows us to integrate with respect to a first; by doing this and againusing the periodicity of g, we get

J=-°-\ e-iM(±Hl )2π Jo V 1 - exu J

\ ( ) d u2π Jo V 1 - exu J

where

b = -Lί* :e- i i 7 n + n + 1 ) ag(a)da .2π Jo

An application of Lemma 1 now completes the proof.The theorem will be proved by expressing Ln in terms of the divided

differences of a certain function q related to / and formed in the points£i> 2, •••>«». We define these formally as follows:

= d(q I zu z,, z3) =

^ = d(q\*u*»

The subscripts on the d's refer to the "order" of the divided differences.In our stochastic model, these expressions as they stand are indeter-minate with probability zero.

By induction it can be shown [13, p. 15] that

(2.6) <2K-i = Σ — 1 / \ »1 <(zk)

868 J. H. CURTISS

which incidentally proves that the divided differences are symmetricfunctions of the zk'&. From (2.1) and (2.6), it is clear that if q = qw(z) =f(z)j(w — z), \w\Φl, then

(2.7) LΛ(f; w) = ωn(w)d(q \zlf , z n ) .

If / is such that its first n — 1 derivatives satisfy Lipschitz con-ditions on I s I = 1, then it is easily shown that the same must be truefor qw{z) =f(z)l(w — z), \w\ Φl. We omit the details.

We need another lemma which will insure that EkLk(f; w) existsfor k = 2, •••, n and can be calculated by interated itegration.

LEMMA 3. If a function f given on \ z | = 1 possesses an (n — 2)thderivative satisfying a Lipschitz condition on |z| = 1, and if dly d2y , dn_1

are respectively the divided differences of f formed successively in thepoints zlfz2f ,zn<m\z\ = 1, then | dx |, | d21, , | dn-λ \ are uniformlybounded for all zlf z2, * ,zn for which these divided differences aredefined.

The proof of this lemma is rather long, and is available elsewhere[5]. With proper completion of the definitions of the divided differencesby continuity, coincident points zk are allowable, but that is of no interestfor present purposes.

Suppose now that Q(z) is any function satisfying the hypotheses ofLemma 2. Let z and t be any two members of the family of randomvariables zlf z2, and let w be any fixed complex number. Then byLemma 2, for any k 0 with z = eia

9 t = eίβ,

(2.8) E2(w - t)(wk+1z~k - z)d(Q \ z, t)

2πJ Jo Jo eia — eίβ

Q ( e ) ( ^ e

) α - 0 - 0 - eiΛ)da2π Jo

- E^w^z-^ί-z)Q{z) .

We now use (2.7) and invoke Fubini's Theorem and Lemma 3 as authorityfor calculating the multiple integral

EnLn(f; w) = Enωn(w)d(q \ zlf , zn)

by integration in any convenient order. In what follows, the operatorEx inside the square brackets refers to integration with respect to zλ

and E2 inside the square brackets refers to double integration withrespect to zx and the zk of the largest subscript. By using (2.8) repeatedly,


we obtain:

EnLn = EnJjl (w - zk)E2(w - zx)(w - zn)d(q \ zu , zn)\Lk=2 J

2π Jo w — ^

1=1 Zx — W

The last expression is the right member of (2.5), and the proof is nowcomplete for | w \ Φ 1.

Because of the singularity of qw{z) = f(z)l(w — z) with | w \ — 1,I s I = 1, the above argument needs further elaboration to establish that(2.3) holds on the unit circle. However we can prove this by a differentapproach. It is well known [13, pp. 24-25] that Ln(f, w) is identicalwith Newton's interpolation formula:

(2.9) Ln(f; w) = ffa) + Σ (w - z,) . . . (w - zk)d(f \ zlf , zk+1) .

Our hypotheses on / insure that the expected value of each term of thisformula exists for all w, and the expected value of the sum (which ofcourse is the sum of the expected values of the terms) is clearly somepolynomial in w defined for all w including | w \ = 1. It is equal to theright member of (2.3) for \w\Φl, and so therefore on | w \ = 1. Thisestablishes (2.3) for all values of w.

An alternative proof of Theorem 1 can be based on (2.9). A con-sequence of our method of proof of Theorem 1 is this:

THEOREM 2. Let f given on \z\ = l be such that En\d(f\zlf ,zn)\exists. Then

(2.10) End{f I *i, , zn) = -A- [*e-«*-1)Θf(eiΘ)dθ2ττ Jo

870 J. H. CURTISS

(2πιht\=i tn

In Theorem 1 the Lipschitz condition on f{n~2) is used only to insurethat I d(q \zu •••,£»)! is integrable. The hypotheses on/ could be replacedby this condition, as we did in Theorem 2, and the restriction on / wouldbe lighter.

A generalization of Theorem 1 to the case in which the unit circleis replaced by an arbitrary Jordan curve C and / is analytic on andinterior to C is discussed in [7]. The probability distribution of thepoints zk on C is defined by the condition that the image points wk = eiθk

under the mapping function z = φ(w) used by Fejer have uniformly andindependently distributed angles θk. The generalization seems unsatis-factory because convergence of EnLn does not take place unless thesingularities of / are all at least a certain distance (characteristic of C)removed from C.

As a result of passing interest here we note that if the points zk

are so distributed, and if / is analytic on and inside a rectifiable Jordancurve C" containing C in its interior, then

ωn(t)

An easy calculation shows that

Γ ffi .ludt,> [wtφ\wt)]n

where t = φ(wt). The Koebe distortion theorem [6, pp. 279-281], suitablymodified for exterior-to-exterior mapping functions, yields the inequality

- I ) 3 < r < R(R + If R > 1 0 < θ < 2 π

f9 9 f

where r is the capacity of C The right member is a decreasing functionof JR, so there exists some value or R such that for all functions / analyticon and inside the level curve C: {z \ z = Φ(Reίθ), R fixed, 0 ^ θ < 2τr},the relation lim^oo 2£»d»_i = 0 holds.

We conclude this section by noting a consequence of Theorem 1which is obtained by combining equation (2.3) with a result due to Walsh[14, pp. 153-154]:

THEOREM 3. Let f be analytic for | z \ < R > 1 but have a singularityon I z I = R. Let Pn(z) be the polynomial of degree at most n — 1 foundby interpolation to f in the nth. roots of unity. Then:

lim [Pn(z) - EnLn(f; z)] = 0

for I # I < R\ uniformly for | z | ^ Rf < R\

3Ψ Equidistributions and uniform, probability distributions^ In thestandard deterministic treatment of polynomial interpolation in the realor complex domain, the interpolation points are presented in a triangularmatrix

(3.1) &: #n

O 2 ! #21 #22

O 3 ! #3χ # 3 2 #33

with the implication that more than one—perhaps all—of the points of thenth row may not have appeared previously. The sequence Ll9 L2, ,Ln, of interpolating polynomials is found by making Ln interpolateto a given function in the points Sn of the nth row. This is the set-upneeded to cover, for example, interpolation in successive sets of equallyspaced points. The stochastic model of the preceding section providesa probabilistic theory for a deterministic interpolation process of thissort if we think of (3.1) as a sample sequence of the stochastic process#i, #2 * with zn a determination of #2; #21 and #22 determinations of #2

and #3; and so forth. This implies that znn is a determination of zn(n+1)ι2.It is convenient now to relabel the random variables zu #2, so as tocorrespond with (3.1). We do this by superscripts, denoting the stochasticprocess now by #n; #21, #22; #31, #32, #23;

We assume once again that the arguments (angles) of the terms znk

are mutually independent and each is uniformly distributed on [0, 2ττ],The Glivenko-Cantelli theorem states that given any sample sequence ofthis process, if Nk(θ) denotes the number of arguments in the first kterms of the sample sequence which do not exceed θ, then with prob-ability one,

(3.2) Bm

for each value of θ. But in the standard deterministic interpolationtheory, a stronger equidistribution property is used [14, pp. 164-166]:Let N*{θ) denote the number of points in the nth set of n points,Sn = {#wi, zn2, , zn J , with arguments not exceeding θ; the requiredcondition is that

(3.3) M ) ±lim*->- n 2π

for each θ. We shall call this the strong equidistribution property for

872 J. H. CURTISS

a sequence such as (3.1). The roots of unity are so distributed. Byelementary methods it is easily shown that if a sequence satisfies (3.3)it satisfies (3.2), but not necessarily conversely. For example if thepoints znk in each of a suitably sparse but infinite set of rows of (3.1)were all equal to a constant α, (3.2) might be true but (3.3) certainlycould not be true.3 It is of interest to ask whether almost every samplesequence of our stochastic process has not only property (3.2), but alsothe strong property (3.3).

The answer is in the affirmative. We here sketch the argument.Let Pr(A) denote the probability of any event A. Consider an infinitesequence of random variables X11; X21, X22; X31, X32, X33; in whichPr(Xnk = l ) = p , Pr(Xnk = 0) = q, q + p = l , fc = l , . . . , w ; n = l, 2 , •••.In this sequence let the random variables in the nth group of n variablesbe mutually independent for n — 1, 2, •••. However, successive groupsof n need not be independent. Let σn = Σΐ=1 Xnkln. In a proof of theStrong Law of Large Numbers for the Bernoulli case given by Feller[8, pp. 190-191], it is shown that for any e > 0 there exists a numberM > 0 constant with respect to n such that

Pr{An:\σn-p\>ε}<^.n2

This means that for any infinite subsequence of the sequence of eventsA19A2, •••, say Anjf j = 1, 2, •••, the series ^ Pr(Anj) converges. Itfollows from the Borel-Cantelli lemma [9, p. 18] [8, p. 188] that theprobability is zero that an infinite number of the events Anj occur.This is the same as saying that Pr(lim^oo σn — p) = 1.

The author is indebted to Professor Kai-Lai Chung for corroboratingthe truth of this result in a letter. Professor Chung fiirst refers to aresult concerning the standard Strong Law of Large Numbers given inhis Columbia University Lecture Notes, 1950-51, in which it is shownthat if Xlf X2, is a sequence of independent identically distributedrandom variables with E(Xj) = 0 and E{Xt) finite, then

X, + X2 + .. + Xn

n>e

converges. He then in the Bernoulli case remarks that the marginalprobability Pr(Bn) — Pr(An) individually for each n, regardless of thejoint probabilities for corresponding collections of the events Bn and An,and this is the essential link between the classical Strong Law and thepresent version.

To show that (3.3) holds with probability one for each Θ, we simply3 The distinction between (3.2) and (3.3) is related to the concept of "well-distributed"

points introduced by Petersen [10].


identify the event Xnk = 1 with the event arg znk ^ θ in the stochasticprocess z11; z21, z22; . However, the Glivenko-Cantelli Theorem sayssomething more with regard to (3.2); namely that (3.2) holds with prob-ability one uniformly in θ [9, p. 20]. Inspection of the proof revealsthat this is true for (3.3) also.

Thus with probability one a sample sequence of the process z11; z21,z22; has the strong equidistribution property uniformly in θl9 and sothis process may properly be considered the stochastic analogue of theequidistributed sequences used in interpolation theory. We note inconclusion that if Ln(f; z) = Ln(f; z \ zlf z2, , zn) in § 2 (see (2.1)) isreplaced by Ln(f; z \ znl, zn2, , znn), then Theorem 1 and Theorem 3 stillare valid, and so is Theorem 2 with znk replacing zk, k — 1, , n, becausethese theorems depend only on the joint probability distributions of then visible random variables.

4* Interpolation in certain strong equidistributions* A number ofyears ago, the late Professor Aurel Wintner asked the author whetherit might be possible to extend Walsh's result (1.1) to the case of inter-polation in equidistributed points on the unit circle, at least for functionsanalytic interior to the circle and satisfying smoothness conditions shortof analyticity on the closed unit disk. Professor Wintner particularlyhad in mind interpolation in the points Sn: ξ, ξ2, , ξn

r where | f | = 1 andξ is no root of unity. It is well known [11, pp. 70-71] that the sequenceSlf S2, is equidistributed in the classical sense (3.2) on | z | = 1.Moreover it is easy to prove by WeyΓs criterion [11, p. 70] that thissequence is strongly equidistributed also if by Sn is meant the nth setof n points in the sequence.

We cannot give an answer here to Wintner's question as to inter-polation in the particular sequence Sn defined above. Theorem 1 aboveseems to be relevant to the more general problem. In the present sectionby combining the roots of unity with a point ξ we shall construct astrongly equidistributed sequence of interpolation points which demon-strates some of the limitations inherent in interpolation in nonequallyspaced points.

In fact, let the nth row of (3.1) consist of «Λjfe=β2ίCί*/(w"lϊ, fc=l, ,w-l,and znn = I, where as above | ξ \ = 1 and ξ is not a root of unity. Thissequence, which we shall denote by S*fS*f f is strongly equidistributed.Let o)n(z) = (z — znl)(z — zn2) (z — znn). It is readily verified that iff(z) is the function 1/z, then for any interpolation points at all,

Here this becomes

874 J. H. CURTISS

L (f z) - i-fl - (a""1

For | z | < l ,

But the value of the right member of (1.1)—that is, the F which appearsin (2.4)—is zero for this function. The perturbation caused by the adjoin-ing of a single extra point to the wth roots of unity transmuted Walsh'sresult. Note that with ξ replaced by ξn in Sn there would have beenno limit at all.

For a general convergence theory for this set of interpolation points,we write Ln in the following form:

k=1 ^n\Znk)\Z %nk)

where d(f \ z, t) is the first divided difference of / formed in z and t,as in § 2. In the present case, ω'n(z) = (n — l)2n~2(2 — | ) + {zn~ι — 1),ωί(««fc) = (n - l)«n*(«»ib - I), ik = 1, , n - 1, ft>;(«wn) = ξ"'1 - 1, and

£.(/;«) -/(f) + {zn~λ ~~l^z ~ ξ) Σ {ll^n^

The redundant factors (1/27Γ) and 27Γ have been inserted to bring outthe fact that the summation is a Riemann sum for the function of θf

d(f\eiθ,ξ).z-eiθ

formed for a partition of [0, 2π] into n — 1 equal parts.We now make the hypothesis that d(f \ eίθ, ξ) is Riemann integrable

with respect to θ. It then follows from the elementary limit theoremsthat for I z \ < 1,

(4.2) \im Ln(f;z)=f(ξ) + ^ -

iίi=i ( ί — «)

d(f\t,ξ)dtt\=i

where F(z) is given by (2.4). The convergence is uniform on any closedsubset of \z\ < 1.

It should be noted that if / is analytic on | z | < 1, continuous on\z\ ^ 1, and such that d(f\t, ξ) is Riemann integrable with respect to

t on 111 = 1, then by Cauchy's Integral Theorem the integral in thelast member of (4.2) is zero and F(z) = f(z), | 2 | <£ 1, so lim^o* Ln(f; z) =f(z), I z I < 1, as in the theory for equally spaced points.

If Sn consists of the (n — 2)th roots of unity plus two distinct"mavericks", ξ± and £2, \ξx | = | | a | = 1 and neither & nor ξ2 a root ofunity, then (4.2) becomes

(4.3) lim Ln{f z) = F(z) - - M«->«> 27ΠJ

Z~^Λ d(f\t,ξuξ2)dt,2πi J ίi=i

in the divided difference notation of §2. It must be assumed that thesecond difference of / formed in the variable point t and the fixed pointsfi and ξ2 is integrable with respect to t. The pattern for adjoiningadditional mavericks to the roots of unity is apparent from (4.3). Nomatter how many mavericks there are, if / is analytic on | z | < 1, con-tinuous on I z I 1, and sufficiently smooth in the respective neighbor-hoods of the mavericks, then lim^oo Ln(f; z) — f{z), \z\ < 1.

This suggests that Professor Wintner's question may have an af-firmative answer, at least for functions which are analytic on \z\ < 1,continuous on | z | S- 1, and infinitely differentiate on | z \ = 1 but notanalytic on | £ | ^ 1 . The question arises as to whether a necessarycondition for an affirmative answer to his question for all strongly equi-distributed interpolation sets, or just for all sets of the type ξ, ξ2, ξ\ ,is t h a t / is analytic on \z\ ^ 1. The sufficiency is of course coveredby Fejer's result.

Recently the author [3] announced some necessary and sufficientconditions for convergent interpolation to functions given on Jordancurves in the complex domain. Some rapid answers can be given bythe sequence S*9 S2*, •••, to certain questions which might arise in con"nection with the role of equidistribution in these necessary and sufficientconditions. The part of the theorem here of interest can be stated asfollows:

(a) Let the rectifiable Jordan curve C contain the origin of itsinterior. A necessary and sufficient condition that

(4.4) lim Ln(f; z) = F(z0) = - L ( _/ί*L dtn—'OT Adi* v J I ε I—-1 1/ <VQ

at a single preselected point z0 of the interior D of C for all continuousf is that the sequence S19 *S2, of interpolation points be such that

(4.5) UmLn(z-*;z0) = 0, fc = 1 , 2 f •••

and at the same time

876 J. H. CURTISS

(4.6)

is uniformly bounded for all n. If (4.6) is not satisfied, then thereexists a continuous f for which lim^oo | Ln(f; z0) | = oo.

(b) If f is analytic on D and continuous on C U D, then (4.6)alone is a sufficient condition for (4.4).

(c) If (4.6) holds at only one point z09 then the image sequenceon the unit circle of S19 S2, under the mapping function used byFejer (§2) is strongly equidistributed.

The first question is whether strong equidistribution with each Sn

containing only distinct points might be sufficient for either (4.5) or (4.6)to hold. We have already seen in (4.1) that when C is the unit circlethe answer for (4.5) is no. As for (4.6), for the set S* the last termof the summation is

zl- 1

The equidistributed points ξ, ξ2, £3, are everywhere dense on | z | = 1,so this term is unbounded for each z0, \zo\ < 1. Therefore (4.6) cannothold for this equidistributed sequence, and furthermore for each z0, \zo\<l,there is a continuous function / for which | Ln(f; z0) | formed in S* isunbounded as n increases.

The second question is whether (4.6) might be necessary as well assufficient for convergent interpolation to all functions / analytic on Dand continuous on C [j D. When C is the unit circle, we have shownthat the sequence S*,S*, ••• provides convergent interpolation to allsuch functions for which d(f\t,ξ) is integrable in t, and this sequencedoes not satisfy (4.6). Thus at least for the Lipschitz subclass of theclass of functions under consideration, (4.6) is not a necessary condition.But the question remains open as to whether if (4.6) is not satisfied, afunction / analytic on D and merely continuous on C U D can alwaysbe constructed such that Ln(f; zQ) —> oz for some zQ9 \zo\ < 1. This ap-pears to be related to another open question communicated to the authorby Professor Philip C. Curtis, Jr.: Given any sequence S19 S2, •••, onC which may or may not satisfy (4.6); can a function / analytic on Dand continuous on C U D always be constructed such that for some pointz0 on C, Lw(/;z 0)^cχ>?

REFERENCES

1. J. H. Curtiss, Interpolation in regularly distributed points, Trans. Amer. Math. Soc,38 (1935), 458-473.


2. , Riemann sums and the fundamental polynomials of Lag range interpolation.Duke Math. J., 8 (1941), 634-646.3. , Interpolation with harmonic and complex polynomials to boundary values,J. of Math, and Mechanics, 9 (1960), 167-192.4. , A stochastic treatment of some classical interpolation problems, Proceedingsof the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Universityof California Press, vol. II, 79-93.5. , Limits and bounds for divided differences on a Jordan curve in the complexdomain, Air Force Technical Note, Contract AF 49(638)-862. AFOSR 1091. To appear inthis Journal.6. P. Dienes, The Taylor Series, Clarendon Press, Oxford, 1931.7. L. Fejer, Interpolation und konforme Abildung, Gottinger Nachrichten, (1918), 319-331.8. W. Feller, An Introduction to Probability Theory and Its Applications, John Wiley,New York, 1957.9. M. Loeve, Probability Theory, Second edition, D. Van Nostrand Co., Inc., Princeton,1960.10. G. M. Petersen, Almost convergence and uniformly distributed sequences, Quart. J.Math., 7 (1956), 188-191.11. G. Pόlya and G. Szegδ, Aufgaben und Lehrsdtze aus der Analysis, Dover Publications,New York, 1945.12. C. D. T. Runge, Theorie und Praxis der Reihen, Leipzig, 1904.13. J. F. Steffensen, Interpolation, Williams and Wilkins, Baltimore, 1927.14. J. L. Walsh, Interpolation and Approximation by Rational Functions in the ComplexDomain, Second edition, Amer. Math. Soc. Colloquium Publications, vol. 20, Providence1956.15. H. Weyl, Uber die Gleichverteilung mod. Eins, Math. Annalen, 17 (1916), 313-352.16. A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 1959.

UNIVERSITY OP MIAMI

HOMOGENEITY OF INFINITE PRODUCTSOF MANIFOLDS WITH BOUNDARY

M. K. FORT, JR.

1. Introduction* In 1931, 0. H. Keller [2] proved that the Hubertcube Q is homogeneous. V. L. Klee, Jr., proved [3] in 1955 that Q ishomogeneous with respect to finite sets, and in 1957 strengthened thisresult [4] by showing that Q is homogeneous with respect to countableclosed sets. Our Theorem 1 extends this latter result to spaces whichare the product of a countably infinite number of manifolds with boundary.Our method of proof exploits the notion of category for the space ofself-homeomorphisms of the product space, and differs considerably fromthe methods of Keller and Klee, who made use of convexity propertiesof linear spaces.

In Theorem 2 we prove that if P is the product of a countablyinfinite number of manifolds with boundary and U and V are countabledense subsets of P, then there is a homeomorphism h of P onto itselfsuch that h\ U] = V. This theorem is analogous to a well known theoremabout Euclidean spaces (see [1"|, p. 44). In a corollary to our Theorem 2,we show that if U is a countable subset of the Hubert cube Q, thenthere is a contraction ht, 0 ^ t ^ 1, on Q such that if 0 < t < 1, thenht is a homeomorphism and ht[Q] Π U — φ.

2. Notation and lemmas. For each positive integer n, we let Mn

be a compact manifold with boundary, and we let Bn be the boundaryof Mn. We let P be the cartesian product space Mx x M2 x M3 xThe projection mapping of P into Mn is denoted by 7ΓW. If x e P, wedenote πn(x) by xn. An admissible metric dn for Mn is chosen so thatMn has diameter less than 2~w, and we then define an admissible metricd for P by letting

d(x, y) = Σ dn(xn9 yn) .

If / and g are mappings on a compact metric space X into a metricspace Y, we let p(f, g) denote the least upper bound of the distancesbetween f(x) and g(x) for x in X.

The set of all homeomorphisms of P onto P is denoted by H.Although the metric space (iJ, p) is not complete, it is topologicallycomplete (i.e. homeomorphic to a complete metric space) and hence is

Received October 3, 1961. Written during a period in which the author was an AlfredP. Sloan Research Fellow. This work was also partially supported by National ScienceFoundation grant NSF-G12972.

879

880 M. K. FORT, JR.

a second category space.The following two lemmas can be proved using standard techniques,

and the proofs are merely outlined.

LEMMA 1. If M is a manifold with boundary B, a is an arclying in B, u and v are the end points of α, and W is an open subsetof M which contains a, then there is a homeomorphism ψ of M ontoM such that ψ(u) — v and ψ(x) = x for x e M — W.

Proof. Let S be the set of all points t of a for which there existsa homeomorphism f o i l onto M such that ψ(u) = t and ψ(x) = x forxe M — W. It is easy to see that S is both open and closed relative to a.

LEMMA 2. If M is a manifold with boundary B, the dimensionof M is at least 2, C is a countable and compact subset of M — B> andφ is a homeomorphism on C into M — B, then φ can be extended to ahomeomorphism Φ on M onto M.

Proof. For each positive integer n, we can obtain compact setsJn and Kn such that:

( i ) C is contained in the interior of Jn and φ[C] is contained inthe interior of Kn;

(ii) each component of Jn and of Kn has diameter less than \\n andis homeomorphic to a spherical ball of dimension equal to that of M;

(iii) for each component D of Jn, φ[D Π C] is contained in a singlecomponent of Kn; and

(iv) Jn 3 Jn+1 and Kn Z) Kn+1.

Using the sets Jn and Kn, it is possible to construct homeomorphismsΦn of M onto M such that:

(i) if D is a component of Jn and Έ is a component of Kn, thenΦn[D] c E if and only if φ]D Π C] c E; and

(ii) Φn+1(x) = Φn(x) for all xeM- Jn.The sequence Φl9 Φ2, ΦB, converges to the desired homeomorphism.

LEMMA 3. // peP, there is a residual subset R of H such thatif heR, then h(p)n e Mn — Bn for each n.

Proof. Let Kn = {h\heH and h(p)ne Bn}. It is obvious that eachKn is closed. We want to prove that Kn if nowhere dense. Thus,suppose he Kn for some n and that ε > 0. We seek ge H — Kn suchthat ρ{g, h) < ε.

Choose an integer m Φ n such that Mm has diameter less than ε.We define M = Mn x Mm. M is also a manifold with boundary, and theboundary B of M is the set (Mn x Bm) U (Bn x Mm). Since heKn, thepoint (h(p)n, h(p)m) is a member of Bn x Mm. Let g be a point of Bm

HOMOGENEITY OF INFINITE PRODUCTS OF MANIFOLDS WITH BOUNDARY 881

such that qΦ h(p)m. There is an arc β in Bn x Mm which joins (h(p)n, h(p)m)to (h(p)n, q) and has diameter less than ε (since Mm has diameter lessthan ε). We may now choose a point re Mn — Bn and an arc 7 joining(r, q) to {h(p)n, q) such that β U 7 is an arc and has diameter less thanε. We let a = β U 7.

We now use Lemma 1 to obtain a homeomorphism ψ of M onto Msuch that ψ maps the point (h(p)n, h(p)m) onto (r, q) and the distancefrom x to ψ(x) is less than ε for all x e M.

Now, we define g e H by letting g{y\ — h(y)k if n Φ k Φ m, andletting

Since #(p)w = r and r$Bn, ge H — Kn. It is easy to see that ρ{g> h) < ε,and hence we have proved that Kn is nowhere dense. We define R =H — \Jn=i Kn. R is a residual set and if h e R, then h(p)n $ Bn for all n.

LEMMA 4. 7/ p and q are points of P> then there is a residualsubset R of H such that if he R, then h(p)n Φ h(q)n for all n.

Proof. We define Jn = {h\heH and h(p)n = h(q)n}. Each Jn isclosed. We want to prove that Jn is nowhere dense. Suppose hejn

and ε > 0. We seek geH — Jn such that ρ{g, h) < ε.It follows from Lemma 3, and the fact that residual subsets of H

are dense in iJ, that there exists feH such that p(f, h) < ε/2 and forall k, f(p)kφBk and f{q\$Bk. If f(p)nΦf(q)n we can let g = /.Otherwise, we choose mΦn so that f(p)mφf(q)m and define M= Mn x Λfw.Since (f(p)n,f(v)J and (f(q)n,f(q)m) are not equal and neither is on theboundary of M", there is a homeomorphis φ of M onto M" such that thedistance from x to 9?(a?) is less than ε/2 for all a? e M and such that thepoints φ((f(p)n, f{p)J) and φ((f(q)n, f(q)m)) have different first coordinates.We now define geH hj letting g(y)k — f(y)k if n Φ k Φ m, and(g(y)n,f(y)J = <P((f(y)n,f(y)m)) It is easy to see that ^(gr,/) < ε/2 andhence ρ(g, h) < ε. Moreover, g(p)n Φ g(q)n and hence ge H — Jn.

We obtain the desired residual set R by letting R = if — UΓ=i «/"*-

THEOREM 1. If A is a closed and countable subset of P and f is ahomeomorphism on A into P, then f can be extended to a homeomorphismF on P onto P.

Proof. There is no loss in generality in assuming that each Mn

has dimension at least 2, for otherwise we could define Sn — M2n-i x M2n

and represent P as S± x S2 x S3 x .It follows from Lemma 3 and Lemma 4 that there is a homeomorphism

882 M. K. FORT, JR.

he H such that for each n, the projection mapping πn maps both h[A]and hf[A] in a one-to-one manner into Mn — Bn. The mapping φn =πjifh^π'1 is one-to-one on πnh[A] onto πnhf[A] and can be extended by-Lemma 2 to a homeomorphism Φn on Mn onto Mn. We obtain 0 e Hby letting 0(#)n = @n(%n)- The desired extension F of / is obtained bydefining i*7 = A " 1 ^ .

Let h be a homeomorphism on a compact space X into a compactspace Γ, and let n be a positive integer. We define

?(Λ, n) = 2~n ίnf {d(h(x), h(y)) \ x, y e X and cZ(α, ») ^ 1/n} .

LEMMA 5. If hly h2, h3, is α sequence of homeomorphisms on Xonto Y such that p(hn, hn+1) < f]{hn, n), then the sequence convergesuniformly to a homeomorphism h on X into Y.

Proof. It is clear that the sequence converges uniformly to acontinuous function h on Xinto Y. We must prove that h is one-to-one.

Suppose u and v are distinct points of X. We choose n > 1 so thatd(u, v) > 1/n. Then, for k ^ n,

d(hk+1(u), hk+1(v)) ^ d(hk(u), hk(v)) - d(hk(u), hk+1(u)) - d(hk(v), hk+1(v))

^ rf(Λfc(u), hk(v)) - 2η{hk, k)

^ d(hk(u)f hk(v)) - 2 2-H(hh{v), h(v))

Thus,

d(h(u)f h(v)) = lim d(hk(u), hk(v))

) ( )

^ d(hn(u), hn(v))l4: , (since n > 1) .

This proves that h is one-to-one and hence a homeomorphism.

THEOREM 2. / / U and V are countable dense subsets of P, thenthere is a homeomorphism h of P onto P such that h[ U] = V.

Proof. As we have remarked in the proof of Theorem 1, there isno loss in generality in assuming that each Mn has dimension at least2. In view of Lemma 3 and Lemma 4, we may also assume that U andV are so situated in P that each πn maps both U and V in a one-to-onemanner into Mn — Bn.

We are going to arrange the points of U and V into sequencesuu u2, u3, and vlf v2, v3, and choose homeomorphisms h{j for all

HOMOGENEITY OF INFINITE PRODUCTS OF MANIFOLDS WITH BOUNDARY 883

positive integers i and j . This is done by a fairly complicated inductiveprocess, the first four steps of which are given below. We let Z7X = Z7,V1 = V, and as soon as ulf , un and vu * ,vn are defined, we letUn+1 = Un- {uu •-,%»}, F w + 1 = Vw - K , vn}. We assume that Uand F are well ordered so as to have the order type of the positiveintegers. We let Hn be the set of homeomorphisms of Mn onto itself.

Step 1. ux \& chosen to be the first point of U and vx is chosen tobe the first point of V. hn e Hx is chosen so that hlλπx{u^ = πx(v^.hu e Hj is the identity for j > 1.

Step 2. v2 is the first point of V2. u2e U2 is chosen near enoughto v2 for us to obtain h21e Hλ so that: p(h21, hn) < τj(hll91) and h^πÛj) —π^Vj) for j = 1, 2. h22 e H2 is chosen so that h22π2{uό) = τra0>i) for i = 1, 2.h2j e Hά is the identity for j > 2.

Step 3. w3 is the first point of U3. v3 e F 3 is chosen near enough to

3 for us to obtain h3i e Hi so that: p(h3i, h2i) < η(h2i, 2) and h^πûi) =^ ) for i = 1, 2 and i = 1, 2, 3. h33eH3 is chosen so that h33π3{u5) =

for i = 1, 2, 3. Λ3i e Hs is the identity for j > 3.

u

4. v4 is the first point of F4. ^4G ?74 is chosen near enoughto v4 for us to obtain hu e H{ so that: p(hu, h3i) < η(h9i9 3) and h^πÛj) —π^Vj) for i = 1, 2, 3 and j = 1, , 4. Λ44 e iϊ4 is chosen so that h^π4{u3) =π4(vj) for j = 1, , 4. A4i e Hά is the identity for j > 4.

We continue this process. By Lemma 5, the homeomorphismshjl9 hj2f fix, converge uniformly to a homeomorphism gό e Hj. It iseasy to see that gfij(Ui) = π^v^ for all i and i . There is determineduniquely a homeomorphism h e H for which TΓ^ = ^TΓ^ for all j . SinceΛ(^) = vt for all i, and U= {uu u2f •}, V= {vlf v2, •}, h is the desiredhomeomorphism.

COROLLARY. If C is a countable subset of the Hilbert cube Q, thenthere is a contraction ht, 0 ^ t ^ 1, defined on Q such that:

( i ) hλ is the identity,(ii) h0 is a constant mapping, and(iii) if 0 < t < 1, Λ* is a homeomorphism of Q into Q and

ht[Q] ΠC = φ.

Proof. We let Mn be the closed interval [—5"~w, 5~w], The resultingspace P may then be thought of as the Hilbert cube Q. (This represen-tation is used since Mn was assumed to have diameter less than 2~n.)We let D be the set of all points x in P such that π^x) is rational for

884 M. K. FORT, JR.

all i, and π^x) = 5"~* for all but a finite number of values of i:Both C U D and D are countable and dense in P, so by Theorem 2

there is a homeomorphism G of P onto P such that G[C U D] = D.We define #t(#) = ί# for 0 ^ t ^ 1 and xe P. Finally, we let fet =G^QtG. It is easy to see that the desired contraction is ht, 0 ^ t ^ 1.

REFERENCES

1. W. Hurewicz and H. Wallman, Dimension Theory, Princeton 1941.2. Ott-Heinrich Keller, Die Homoiomorphie der kompakten konvexen Mengen im HilbertschenRaum, Math. Ann., 105 (1931), 748-758.3. V. L. Klee, Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc,7 8 (1955), 30-45.4. , Homogeneity of infinite-dimensional parallelotopes, Annals of Math., 66 (1957),454-460.

UNIVERSITY OF GEORGIA

FAMILIES OF INDUCED REPRESENTATIONS

JAMES GLIMM

In [11], Mackey constructed certain representations (the inducedrepresentations) of a group G. If the group is acting on a measurespace X then the construction also gives a projection valued measure Pon X which is a system of imprimitivity for the representation UoίG.(P(σE) — U{σ)P(E)U{σ~1).) In this paper we determine the topologyin the set of equivalence classes of induced pairs U, P whose joint actionis irreducible, provided certain restrictions are imposed on G and X.This set of pairs is (homeomorphic to) a space W/G of orbits, whereW consists of fibers over X as a base space and G acts on W. Thefiber over x is Gx, the space of equivalence classes of irreducible repre-sentations of Gx = {7: jx = x). The principal restriction on G and X isequivalent to assuming that Gx is a continuous function of x. (See theAppendix.) One might hope that in interesting cases X could be ex-pressed as a finite disjoint union of subsets upon which our assumptionsare satisfied.

One of the motivations for this paper was the hope of introducingin certain cases a differentiate or real analytic structure into W/G. Ifif is a manifold (except perhaps for a set of singular points), if G isan analytic group and if G acts smoothly on W then W\G is a manifold,except perhaps for a set of singular points, if W\G is countably sepa-rated (if there are Borel sets Wlf W2, in W which are G invariantand which separate points of WIG). This is a simple consequence of[14, Theorem 8, page 19] and [6, Theorem 1] and does not depend uponthe special nature of W. In particular it applies equally well to a closedsubset K of W which is a manifold and upon which G acts smoothly.As might be expected, K\G being countably separated is equivalent toall representations of a certain C*-algebra being of type I. The as-sumption that W is a manifold except for singular points is unsatis-factory. One would like to assume that X is a manifold and that Gacts on X smoothly and conclude that W is a manifold (except perhapsfor singular points) if all the Gx are type I groups. Whether this istrue is not known even when X is a point. The results of this paperpresumably have implications for the representations of analytic groupswhich have closed normal subgroups.

The group G and the topological space X considered in the paperwill be assumed to satisfy the second axiom of countability. This isnot used until § 2 and in view of [10, 1], it would not be surprising

Received November 22, 1961. This work was supported in part by a grant from theOSR and in part by a grant from the National Science Foundation, NSF G-19684.

885

886 JAMES GLIMM

if Theorem 2.1 were true without this assumption. That φ is a repre-sentation of a group (resp. * algebra ££) means that the representationspace ξ>(φ) is a Hubert space and that φ is a unitary representation(resp. * representation and φ{β)§(φ) is dense in £>(<£>)). For any locallycompact space Y, C0(Y) denotes the set of complex valued continuousfunctions on Y with compact support.

1. Group algebras. In this section we study *-algebras which arefields of group algebras and which are associated with a locally compactgroup G acting as a topological transformation group on a locally com-pact T2 space X. That G is a topological transformation group meansthat there is a jointly continuous map (7, x) —• yx from G x X into Xsuch that {β~xy)x = β~\^x) and ex — x. Suppose a left invariant Haarmeasure d(x, σ) — dσ can be chosen on the isotropy subgroups Gx "con-tinuously," that is so that for each/in C0(G), the function x—>l f(σ)dσ

defined on X is continuous. Let Y = {(a?, σ): x e X and σ e Gx). ThenF is a closed subspace of I x G and so is locally compact.

The continuity requirement of the Haar measures could also beexpressed by saying that x —> d{x, σ) is a w*-continuous map from X toregular Borel measures on G.

LEMMA 1.1. Let x —• dμ(x9 σ) be a w*-continuous map from X to

the regular Borel measures on G. For each compact subset K of X x G

there is a constant M = M{K) such that \\f{x, σ)dμ(x, σ) ^MWfW*

for all f in C0(K) and x in X.There are compact subsets Kλ and K2 of X and G respectively such

that Kc. Kx x K2. If g e C0(G) and g = 1 on K2, let M be the supremum

of ί| g(σ) I dμ(x, σ) as x varies in Kλ. life CQ(K) then \f(x, σ)dμ{x, σ)j 1 J

is dominated by \\f\\oo\\g{σ)\dμ{x,σ) ^ ||/||<Jfcf if xeK± and is equal to

zero if x 0 JBLΊ.

It follows from Lemma 1.1 that \ f(x, σ)dσ is a jointly continuousJβx

function of / in C0(K) and x in X.Let Jα be the modular function for Gx, d{x, στ) = ώ(#, α )Λ(^) F o r

a suitably chosen / in C0(G),

f(σ)dσx

and so as a function on Y, Ax{τ) is continuous. If f,geCQ{Y) define

= ( f(x, p)g(x, p-χσJGx

FAMILIES OF INDUCED REPRESENTATIONS 887

Then /*# and / * e CQ(Y) and C0(Y) is a *-algebra. It is also an algebraof vector fields defined on Xand having values in the CO(GX) If fe C0(Y),

let 11 /1 |i = supx€X I \f(x,σ)\ dσ and let 11 /11 be the supremum of 11 <p(f) 11,Jθx

for φ a representation of C0(Y) which is continuous in the inductivelimit topology on C0(Y) (the topology which is the inductive limit ofthe uniform topologies on the C0(K) for K compact). The next lemmashows that | | / | | < °°. It then follows that the completion ί£ of C0(Y)in || || is a C*-algebra.

LEMMA 1.1 A1. || II = II * Ili If φ is an irreducible representationof $t then there is a unique x in X and a unique representation φx

of Gx such that

and x is determined uniquely by the kernel of φ. Furthermore & isclosed under multiplication by bounded continuous functions on X.

Let φ be a continuous irreducible representation of C0(Y) on aHubert space ξ>. Let XQ = {x: x e X and for some neighborhood Nx ofx, kernel φ contains all/in C0(Y) which vanish off Nx (or more precisely,off (Nx x G) n Y)}. Then Xo Φ X. If x and y are distinct elementsof X ~ XQ then there are disjoint neighborhoods Nx and Ny of x and yrespectively and elements fx and fy of Co( Y) ~ kernel φ which vanishoff Nx and Nυ respectively. Then φ(C0(Y))φ(fβ)§ and φ(C0(7)M/,)§are orthogonal nonzero invariant subspaces of ξ>. This contradicts theirreducibility of ψ and so XQ = X ~ {x} for some x. It is now evidentfrom the definition of Xo that if f{x, •) = 0 then / e kernel φ. Hencethere is a representation φx of CO(GX) for which <p(f) = φx{f{x, •))> a n ( ione can check that φx is continuous. Thus φx comes from a represen-tation, also called φx, of Gx and this implies | | φ ( / ) | | ^ \θx\f(%90)\dσ.

The first two statements of the lemma follow immediately. If A, is abounded continuous function on X then || φ(hf) || = | h(x) | || φ(f) || ^||AΊ|oo||/||, and so multiplication by h is an operator on C0(Y) which iscontinuous in 11 11. It thus has a unique continuous extension to all ofS. If we regard ffi as functions from X to the C* -group algebras ofthe Gx then this extension of multiplication by h is still multiplicationby h.

If fe C0(Gy-lχ) then the functional

defines a left invariant integral on Gy-lχ. Thus there exists a uniquepositive number c(x, 7) for which

1 This is based in part upon a lemma supplied by R. Blattner.

888 JAMES GLIMM

(1.1) c(x,y)[ f(y1σy)dσ = [ f(σ)dσ .

If we choose / to be a nonnegative element of C0(G) which is positiveat e then (1.1) implies that c(x, y) is jointly continuous in x and 7. Itis easy to see that the identities

c(x, βy) = c(x, βWβ^x, y)

c(x, τ) - Δx(τ) c(%, e) = 1

are true for β,yeG,τeGx. Also ^ ^ ( 7 ^ 7 ) = Δx{τ) since if / is asuitable element of C0(Gy-lχ) then

f(y~1σy)dσ

X

= ( /(σy-'τ-^dσ/l f(σ)dσJOy~lX I JGy-lX

= Δy_lχ{y~ιτy) .

PROPOSITION 1.2. // / G C O (Γ) then yκ(f) e C0(Y), where

, σ) = f(y~ιx> y~xσy)c(x, 7) .

7^ fcαs α unique extension to an automorphism yκ of & and 7 —> yκ isa strongly continuous representation of G on $.

There is no difficulty in seeing that yκ(f)eC0(Y). If f,geC0(Y)then

, σ) = \ f(y"% PMy'1^, p~~ιy~λσy)c(xf y)dρjGy-lx

, yfdp

σ) = /*(τ"1», y^y)c(xf 7)

, 7)

(», σ)

and yκ is an automorphism of C0(Y). yκ is continuous in the inductivelimit topology and so φoyκ is a continuous representation of C0(Y) ifψ is. yκ is thus continuous in || ||. Hence it has a unique continuousextension to ®, and the extension is an automorphism. Also

so 7—*yκ is a representation. If fe C0(Y) and 7-*70 then yκ{f)—>yQK{f)uniformly with support contained in a fixed compact set and so in the

norm || ||. It follows that yκ is strongly continuous.

G acts on the dual I of ffi as a topological transformation group,in fact more generally we have the following lemma; we do not claimthat this result is original.

LEMMA 1.3. Let SI be a C*-algebra with dual SI and let there be

a strongly continuous representation of a topological group G as auto-

morphisms of 21. Then the map (7, φ)—^yφ — φoy-1 from G x 21 into

21 makes G into a topological transformation group acting on 21.

21 is the set of equivalence classes of irreducible representations of21 with the hull kernel topology, which is the topology which has as asubbasis for closed sets the sets of the form {φ: kernel φ ZD $} where3 is an ideal (closed two sided) in 21. It is evident that (β~λy)φ —β'^yφ) and that y{φ: kernel φ Ί) $} = {φ y'1: kernel φ 3 $} = {φ7~1(kernel φ) z) $} = {φ: kernel φ D 7$} so each 7 in G acts by homeo-morphisms of 31. Thus we have only to show the joint continuity ofthe map (7, φ) —>Ίψ at 7 = e. A subbasic neighborhood of φ is givenby N = {ψ: kernel f J S 1 where $5 is an ideal which is not containedin kernel φ. There is a positive A in $ which is not in kernel φ, byLemma 2.3 of [16]. Let M = {ψ: \\ f(A) \\ > || <p(A) ||/2}. Let / be acontinuous function which is zero on [0, || φ(A) ||/2] and positive elsewhere.M is open since M = {ψ: ψ(f{A)) Φ 0}. For all 7 sufficiently near β,II Ί~\A) - A II < II φ(A) H/2 and for such 7 and for ψ in M, \\ψ- 7~1(A) || >0 so jψ 6 N and the proof is complete.

If Z is the structure space of SI (the set of kernels of irreduciblerepresentations of 21) with the hull kernel topology then the map (7, z) —*72 = {y(A): Aez} form G x Z into Z makes G into a topological trans-formation group on Z. This follows from Lemma 1.3 and from the factsthat 7 kernel φ — kernel yφ and that φ —> kernel φ is an open continuousmap of 31 onto Z.

Let Z be the structure space of £ϊ, let φ be a representation of G.By a system of imprimitivity for φ based on X (resp. Z) we mean aregular countably additive projection valued measure P defined on theBorel subsets of X (resp. Z) with values acting on φ(<p) such that P(X)(resp. P(Z)) = I and ^(7)P(£r)^)(7"1) = P(yE) for all 7 in G and allBorel sets E in X (resp. Z), cf. [11]. We shall call the pair (<p9 P) arepresentation of G, X (resp. G, Z). Here the Borel sets are the elementsof the smallest σ-ring containing the open sets and regular means thatfor open U, P(U) = V {P(C): C is a compact Borel set contained in £/}.

There is a *-algebra associated with representations of G, X. It isIhe set C0(X x G) with multiplication and involution defined by

890 JAMES GLIMM

(1.2) f*g(x, 7) = ( fix, β)g(β-% β~~λΊ)dβJG

(1.3) f*(χ, 7) = f(r-% 7-1)

for f,ge C0(X x G), dβ a left invariant Haar measure and Δ the modularfunction (dβj = Δ(j)dβ) of G. This definition is essentially that of [2,p. 310]. There is also a multiplication between elements / of C0(Y}(resp. C0(X), C0(G)) and elements gr of C0(X x G) given by

(1.4) /*flf(a?, 7) = ί /(», (J)flf(a?, α-1

(1.5) fg(χ,Ύ)=f(χ)g(χ,Ύ)

(1.6) /*flf(a;, 7) =

and there is a norm on C0(X x G) given by

(1.7) IΓf/lli=

THEOREM 1.4. C0(X x G) is a normed *-algebra with multiplication,,involution and norm given by (1.2), (1.3) and (1.7) respectively andaddition and scalar multiplication defined pointwise; involution isisometric. It is also an algebra over the ring C0(Y) (resp. G0(X), C0(G))with scalar multiplication given by (1.4) (resp. 1.5), 1.6)).

THEOREM 1.5. There is a one-to-one correspondence between bounded(in || 111) representations φ0 of C0(X x G) and representations (φ, P) ofG, X. The representation φ0 which corresponds to φ, P is given by

(1.8) φo(f) = ( ί f(x,

The images of φ0 and of the corresponding (φ, P) generate the samevon Neumann algebra. φ0 is norm decreasing (||<po(/)ll ^ ll/lli) Aunitary operator implements an equivalence between representations ψ,P and φ', Pf of G, X if and only if it implements an equivalencebetween the corresponding φ0 and ψ[.

THEOREM 1.6. There is a "canonical procedure" for extendingrepresentations (φ, P) of G, X to representations (φ, R) of G, Z.

If ze Z, let φ be an irreducible representation of $ with kernel z.Let x = π(z) be the x determined by Lemma 1.1A. If E is a closedsubset of X then π~\E) = {z:f& c z if f{E) = 0, / e C0(X)} and is closed.Thus π is continuous and π~\E) is a Borel set if E is. That R extendsP means that R(π-\E)) = P(E) for all Borel subsets E of X.


Proof of Theorem 1.4. Let / and g be in C0{X x G). Then

and

(f*g)*(χ,Ύ) =

= \ g(β-*χ, β-Yάiβ-1)/^, Ί'1β)-Δ{Ί-1β)d

JG

= \ g*(x, β)f*(β-% β~ΎΊ)dβ = (flf* */*)(&, 7)

and (1.3) defines an involution. Suppose that x —• d/ (x, 7) is a functionfrom X to the finite measures on G which is w*-continuous and is suchthat \JxEχ support dμ(%, 7) is contained a compact set. If fe CQ(X x G),define μ*f by the formula

μ*f(x, 7) - , β) .

Then /1*/ has compact support, and by Lemma 1.1, μ*feC0{X x G).Furthermore

(μ*(f*9))(x, 7) = xf a)

^x, β-^dβdμix, a)

= ((μ*f)*g)(χ, 7)

In particular if d/i(x, 7) = &(#, 7)^7, fe 6 C0(X x G) then this proves thatmultiplication is associative. If fex and fe2 are in C0(Y), then the casedμ(xf σ) = }φ}9 σ)[Ax{σ)IΔ(σ)Yl2d(x, σ) proves that h1^(f*g) = (fei*/)*flr.Let'ω(x, σ) = [Jx(σ)/J(σ)]1/2. The formula ht*{h2*g) = (h1*h2)*g followsfrom the associative law in the measure algebra of G and the fact thatω(fe!*fe2) == {ωh^)*{ωh^. The remaining algebraic assertions of Theorem1.4 are easy to verify.

The function sup {| g(x, 7) I : x e X} is a lower semicontinuous functionof 7 and so is measurable. It is bounded and has compact support andso is integrable. If /, g e C0(X x (?)

dy\\f*g Hi = ( sup I ( /(α?, β)g(β-% β'ιΊ)dβJG xβX I JG«

g f ( sup|/(» f β) I sup I g(β-% β-'-r) I dβdj =jGJG xex xex

892 JAMES GLIMM

LEMMA 1.7.2 Let 2ί be a normed *-algebra, let 33 be a *-algebrcirand let θ be a representation of 33 as bounded operators on 21 such thata*(θ(b)a2) = (θ(b*)aύ*a2 for al9 a2 in 21 and b in 33. Let φ be a con-tinuous representation of 21. Then there is a unique representationψ of S3 such that

(1.9) f(b)φ(a) = φ{θ{b)a)

for a in % and b in S3. Moreover \\ψ{b)\\ S 11 #(&*&) ||1/2 and f(33) iscontained in the weak closure of

There is at most one representation ψ satisfying (1.9). If A! com-mutes with φ(2X) then A'ψ(b)φ(a) = ψ(b)φ(a)A' = ψ{b)Arφ{a) and A! com-mutes with i/r(33). By the double commutant theorem, ψ(^&) is in theweak closure of ^(21).

To prove the existence of ψ(b) it is sufficient to consider the casewhere the representation space ξ> of φ has a vector x which is cyclicwith respect to ^(2ί). Let a be in 21, b be in S3. Then

|| φ(θ(b)a)x || = {φ{{θ{b)aYΘ{b)a)x, x)^

= (φ(α*#(δ*&)α)x, x)112

= {φ{θ{b*b)a)x, φ(a)x)112

^ \\φ(θ(b*b)a)x\\ll2\\φ(a)x\\112 .

Iterating this inequality, we have

II φ{θ(b)a)x || ^ || ^(&*6)an-1α)a? ||2"w || ^(α)a? H1"2""

^ II φ \Γn II ^ * & ) ll1/2 II a \\2'n \\ x \Γn \\ φ{a)χ \r2~n,

and taking limits, \\φ{θ{b)a)x\\ || θ(b*b) | | 1 / 2 1| φ(a)x ||. Thus (1.9) is anunambiguous definition of ^(6) on φ($ί)x, ψ(b) is bounded and has aunique bounded extension, ^(6), defined on all of ξ).

Formula (1.9) shows that ψ is linear and multiplicative. ψ(b)* —

is dense in ξ) since #(33)21 is dense in 2ί, since φ is bounded andsince ^(2I)ξ> is dense in φ. Thus ψ is a representation and the proofis complete.

Proof of Theorem 1.5. The integral I fix, i)dP(x) is the ordinaryJx

uniformly convergent spectral integral; it is by definition the uniformlimit of approximating sums Σ?=i P(^ί)f(χn 7), where X is a disjointunion of the Borel sets E19 •• ,£ r

w and Xi^E^ Since / is continuous2 We are indebted to R. Blattner for this lemma and its proof. This replaced consider-

ably more complicated arguments, some of which were in the spirit of [13, §5 and 6] andappeared to be limited to separable situations.

and has compact support, the integral I f(x, j)dP(x) exists and is aJX

continuous function (in the operator norm) of 7 with compact support.Thus φo(f) exists; \\φo(f)\\ ^ II/Hi follows from the fact that

f(x,-y)dP(x)

To show that φ0 is a representation, let / and g be in C0(X x G)and let p and q be in lQ(φ). Then

(<Po(f*9)p, q) = \ (\ \ /(«, β)g(β~1xf β'1y)dβdP(x)φ('y)pf q)djJG\JXJG

= \ lim tjGf [E1,'",En] ΐ=l

= ( f lim Σ (PiEdfixt, β)g(β-1χi, β~xΊ)φ{Ί)p, g)dydβJG JG {Elt ,En} i = l

= ί ί lim Σ (P(E{)f(x(, β)φ(β) Σ P(β-1Ei)giβ-1xi, j)φ(j)p, q)djdβJG JG {E1,'",En} i=l 3=1

= \\(\ f(x,β)dP(x)φ(β)\ g(x,y)dP(x)φ(y)p,q)dydβJGJGKJX JX

= ((Po(f)(Po(9)Pf Q)

and

(Φo(f*)P, Q) = SXSX^(7"^^ ^"^(T^dPίίcMTjp, ^)dτ

= 5β(Jχ/(^» ^-cίP^MT-1)^, q)dy

= \ (p,φ(i)\ f(ΎX,l)dP(x)p)dyJG\ JX I

= \ (p,\ f(%, Ύ)dP(x)φ(y)q)dy = (p, φo(f)q)JG\ JX /

since φ(y) \ h(fyx)dP(x)φ(y~1) = \ h(x)dP(x) for any h in C0(X), as isjx t t Jx

seen by considering approximating sums to the spectral integrals. Leth be in C0(G) with support K, and let hn be a net in C0(X) which eventu-ally has the value one on each compact subset of X, and suppose 0 ^hn ^ 1. Then 1 hn(x)dP(x) converges strongly to / and so

ί hn(x)dP(x)φ(Ύ)pJx

converges to φ(y)p uniformly for all 7 in K. Thus

I (<Po(Kh)p - φ(h)p, q) I= I \j\x

h

894 JAMES GLIMM

(If II f^ s u p I h(y) I s u p hn(x)dP(x)φ(y)p - φ(y)p\\ \\q\\\ dyyeκ

and so φo(hnh)p—>φ(h)p strongly. This proves that the set φQ(C0(XxG))&(φ)is dense in $(<p) and since φ0 is linear, it is a representation. Since theintegrals with respect to dP and dy are weak limits of approximatingsums, φo(Co(X x G)) lies in the von Neumann algebra generated by theimages of φ and P. We have also proved that φ(C0(G)) (and so φ{G))lies in the weak closure of φQ(C0(X x G)).

Suppose we are given a representation ψ0 of C0(X x G) which iscontinuous in || Ili ϊ n Lemma 1.7 let S3 be the algebra C0(X) (resp.C0(G)) and let θ be the multiplication defined by (1.5) (resp. 1.6)). Ife,fe C0(X xG),ge C0(X) and h e C0(G) then

e**(gf){x,i) =

and e* *(&*/) = (&**β)**/. To prove the latter formula one could eithercompute the integrals in question or, as is easier, observe that theformula is true for h in C0(X x G) and then approximate h in C0(G) byelements of C0(X x G). Moreover | | 0 | | ^ 1 in both cases. By Lemma1.7 there are representations ψ of CQ(G) and ^ x of CQ(X) such thatΨi(Q)ψo(f) = ΨoiQf), Ψ(h)fo(f) = φo(h*f). Since ψ is continuous it comesfrom a representation ψ of G, and f(l)f(h) = ^(fe(7"le))- I f w e l e t h

run through an approximate identity and use the formula h(y~1 )*f{x9a) =h^f(y-ιxfy-1a)J we conclude that ^(7)^0(/) = Ψo(/(7"1 , Ύ'1 •))•implies Ψ(y)f1(g)ψo(f) = t i ^ T " 1 ))toί/ίT"1 , 7"1 •)) = ^i(ί7(7"and ^(7)^1(^)n/r(7~1) = Th(0(7~le))« By standard methods (compare [9, p.93, Theorem], [7, p. 239, Theorem D], or Theorem 1.9), ψx can be ex-tended uniquely to a regular countably additive projection valued measureP on X. Let KE be the characteristic function of a Borel set E. SinceKAr1 •) = ^ ( 0 , t(y)P(E)f (y-1) = P(7-Er) and (α/r, P) is a representationof (G, X). It follows from Lemma 1.7 that ψ(C0(X)) is contained inthe weak closure of ψo(CQ(X x G)) and by monotone limits, this is alsotrue for the range of P.

Let <p0 be defined by (1.8) (with ψ replaced by φ), let feC0(X),g e C0(G), h e C0(X x G). Then fg e C0(X x G) and the finite linear com-binations of such elements of C0(X x G) are dense in C0(X x G). Ifq,re φo(Co(X x G)Mf0) then

(<Po(fg)1ro(h)q, r) = f^^f(x)g(y)dP(x)f(y)dyf0{h)q, r)


= ί (Ψi(f)θ(y)ψ(Ύ)Ψo(h)q9 r)dyJG

, r)

= (Ψo(fg)fo(h)q, r)

and so φo~ ψ0. Thus the correspondence defined by (1. 8) is onto fromrepresentations of G, X to representations of CQ(X x (?); one can alsocheck that it is one-to-one. The statement concerning unitary equivalenceis verified by a direct computation.

THEOREM 1.8. // φ, P is a representation of G, X then the formula

where fe C0(Y), g e C0(X x G) and φ0 is defined by Theorem 1.5, definesa representation φ1 of ffi. The image of φλ lies in the von Neumannalgebra generated by the images of φ and P.

Let the Sί (resp. S3) in Lemma 1.7 be C0(X x G) (resp. C0(Y)) andlet θ be the multiplication defined by (1.4). Let e, g be in C0(X x G)and let / be in C0(Y). Then

β**(/*flr)(α?,7)

\ c(x1β)e(β-%β-l

• gφ^x, β-1σ-1Ί)[Δx{σ)Δ{σ-1)]1^dσdβ

ί ( c(x, β)e(β-1x, β-'JGJGx

ί ( efβ-% σβ-'YΔiβ-^fφ-'x, σ)

• g(β-% β-17)[Δβ_lχ(σ-')J(

( t Piβ-'x, σ-ψeiβ-'x, σβ-1)-Δψ-1)g{β-1x,

β~lx

896 JAMES GLIMM

( f ( , )JG

= (f**e)**g(x,y) ,

ll/*ff Ik ^ ( sup [ \f(x, σ)g(x, σ^)[Δx{σ)Δ{σ-')}^ \ dσdi

and

= sup ( ( \f(x, σ)g(x, σ-17)[z/,(σ)//(σ-1)]1'21 dσdyx JOJQX

rsince the function 7—> \ \f{x,σ)g(x,σ~1y)\dσ is continuous and has

JθX

compact support for each x in X. We apply Fubini's theorem, substituteΎ—>σy, and conclude that

(l.ii) \\f*g\\i £ | | / ( x , ^ ) M » 4 O ] 1 / 2 H i l k l l i .

Lemma 1.7 shows that (1.10) defines a representation of C0(Y) and Lemma1.1, the bound in 1.11) and Lemma 1.7 show that φx is continuous inthe inductive limit topology on C0(Y). By the definition of || |l><Pi iscontinuous in || || and defines a representation of $.

Let S be the completion of CQ(X x G) in the norm | | / | | = sup {||<£>(/)!|:φ is a representation of C0(X x G) which is continuous in || HJ. Then8 is a C*-algebra. It follows from Theorem 1.8 that the multiplicationdefined by (1.4) extends to a multiplication between & and 8.

THEOREM 1.9. Let ψ be a representation of a C*-algebra & andlet Z be the structure space of S. If U is an open Borel subset of Zt

let R(U) be the projection onto the closed span of

Then R can be extended uniquely to a countably additive projectionvalued measure on the Borel subsets of Z. The image of R is containedin the center of the weak closure of

Let Sf be the set of proper differences of open sets and let & bethe set of finite disjoint unions of elements of 3f. By [7, §5, exercise(2) and (3)], & is a ring and by [7, §6, Theorem B] & is the smallestclass of sets containing <% and closed under sequential monotone limits.Thus R has at most one extension to a projection valued Borel measureon Z. & is the class of Borel sets.

We extend R to Sf. Let Όλ = Eλ~ Fλ and D2 = E2 - F2 be in 3fwhere E{ and Ft are open and Ei Z) Ft and suppose D1'D D2. We assertthat R{Eλ) - R(F,) ^ R(E2) - R(F2). If z e Z and / e St9 let f(z) be the

element / + z in the C*-algebra S£/s. Then fef){z:zeZ~ U} if andonly if f(z) = 0 for all z not in U, and in this case we say that /vanishes off U and we let $( U) denote the set of all / in fi whichvanish off U. Let p be in Range R{Fλ) and let g be in Range R(E2) —R(F2). If fe& and / vanishes off F2 then f(/)g - 0 and q (resp. p)can be approximated by vectors of the form ψ{g)q (resp. ψ(h)p) whereg (resp. h) vanishes off E2 (resp. FJ. Then (p, q) can be approximatedby (p, ψ(h*g)q) which is zero since h*g = 0 off E2 (Ί F1d F2. ThusR{Fλ) _L J?(#2) - R(F2). QίCEί) + $(F 2) is an ideal contained in ^(E, U F2)and its closure $ is equal to $(£Ί U F2) since otherwise $(2?i U F2) hasan irreducible representation φ which annihilates $, 3ί(£72). Thus g can be approximated byelements fλ + / 2 of ί£, with /x in $(£Ί) and /2 in ^(i^), and q can beapproximated by ψif^q + ψ(f2)q = ψ(fi)Q. This proves that <? 6 RangeΛCEy, Λ ί ^ ) ^ R(E2) - R(F2) and Λ(,Ei) - RiFJ ^ i ? ^ ) - R(F2). IfA = A then R{Eλ) - R(Fλ) = R(E2) - R{F2), and R{D) is defined unam-biguously by the formula R(D) = ^ ( A ) - R{FX).

Let Dλ = Eτ~ Fλ and D2 = E2 ~ F2 be in ^ , where ^ =) andEi and ^ are open and suppose Dx Π D2 = <ρ. Let p be in Range i?(A)and let q be in Range R(D2). Then p (resp. g) can be approximatedby ψ{f)p (resp. ψ(g)q) where /(resp #) vanishes off Eλ (resp. i?2). flf*/vanishes off Eλ C\ E2ci Fx{j F2 and so g*f can be approximated by ele-ments hλ + h2 of S with /^ vanishing off i^. Thus (p, q) can be ap-proximated by (ψ(g*f)p, q) and by (ψih^p + ψ(h2)p, q), which is zero.This proves that E(A) 1 # ( A ) .

We prove that ϋ is countably additive on £&. Let D and Dif

i = 1, , 00, be in ^ , let £> = Ϊ7 - i 7 and A = E, - ^ where Ϊ7 z> F,A D i^ and E, F, E{ and F{ are open and suppose D = \J?=1 A andsuppose the A ' s are disjoint. Then R(D) ^ i2(A) and i2(ΰ) ^ ΣΓ=i Λ(A).To prove R(D) = ΣΓ=i -B(A) w e assume the contrary and we supposewithout loss of generality that A = 0 = A , Ex = E = F± andE2 — F — F2. Let λx, λ2 and λ3 be real continuous functions suchthat 0 g λ< ^ 1, \i(0) = 0, λ<(l) = 1, λxλ2 = λ2, λ2λ3 = λ3, andXi(x) > 0 if xe[l/2, 1]. If ge®, if 0 ^ ^ ^ / , if peξ>(f) and ifII t(ff)3> - V II ^ II V11/3 then f ( λ 3 ( # ^ 0. In fact if f(X3(g)P = 0 andif P is the spectral projection for ψ(g) associated with the interval [1/2,1] then Pp = 0 and ||ψ(g)p|| ^ \\p||/2 and | | f (^)p-p\\ ^ | |p| |/2. There isby assumption a nonzero p in Range R(D) — H i ^ ( A ) . We can choosea g in ίϊ which vanishes off Ex so that ^ = ψ(λ3(g))p Φ 0. Let hλ =\(g), let gx = \(g). Let w be a positive integer and suppose inductivelythat we have chosen

898 JAMES GLIMM

(a) gn in ffi

(b) nonzero vectors pu •••, pn in Range R(D) —

(c) Λy in ^(Fj) whenever p3 e Range

in such a manner that iί j ^ k ^ n then

( i ) P j f JL Range R{E0) =φ p* J- Range

(ii) p, 6 Range R{F3) =>pke Range i 2 ( ^ ) and ψ(h3)pk = pΛ

(iii) p y G Range #(2^), p* G Range R{Fk), and i < k 4> fe^ = feΛ

(iv) 0 ^ h j I; 0 ^ gn^ I,

and if i is the largest index for which p{ e Range R(Fi) and if i ^ k ^ nthen

(v) h,gn = gn and f(gn)pk = pk.

If (J - R(En+1))pn ψ 0, let pn+1 = (J - R(En+1))pn and let flrn+1 = </π.For each C in ^ , Range R(C) is invariant under φ(®), and since ^(ίϊ)is closed under the taking of adjoints, R(C) commutes with ψ(!&). R(C)is also a weak limit point of φ(&) and so R(C) is in the center of f(!&)~,the weak closure of ψ(!&). Using this, it is easy to see that the inductiveassumptions are satisfied for n + 1. If (I — R(En+1))pn = 0 then 0 ΦR(Fn+1)pn = f(gn)R(Fn+1)pn. Thus there is a g in ® which vanishes offi^n+1 such that p w + 1 = ψ(MQn99n))R(Fn+1)pn Φ 0. Let Λn+1 = \(gnggn) andlet 0Λ+1 = X2(gn99n). Since Xk(gnggn) is a limit of polynomials in 0n0flrn,fcifen+i = fen+1, and the remaining inductive assumptions are easy to verify.

Let 3)ϊ be the linear subspace of & + XI generated by / and hά ifPi e Range R{F3) and ( ^ ( ^ ) if ps ± Range iϊ(£?y), j = 1, 2, . . Let ftbe the linear functional on 2Dΐ defined by po(I) = 1, ρQ(h3) = 1 if p3- e RangeR(Fj) and po(3(Ed)) = 0 if p, _L Range i 2 ( ^ ). This definition is consistantand p0 is a state ( = positive linear functional normalized by po(I) = 1)of 2W, since ρQ = (limn ω^o i/r/|| pw ||2) | 3ft, where ωVn is the linear functionalA —* (ApΛ, pw) defined on operators on φ(^). ft is an extreme point ofthe set of states of 3ft. In fact let pQ = aτx + (1 — α)r2, with a e (0,1]and τx and τ2 states. Since ^s(Ej) is generated by its positive elements[16, Lemma 2.3], τffiiEj)) = 0 if p, _L Range R(E3). If p, G Range R{F3)then Tiίfe. ) ^ 1 and 1 = aτ^hj) + (1 - ^)τ2(fey) ^a + l - a = l. Thusthere is equality throughout and τ^hj) = 1, τx = ft, and ft is an extremepoint, ft can be extended to a state p of ® + λ l by a Hahn-Banachtype argument and applying the Krein Milman Theorem to the set ofsuch extensions, it is possible to choose p to be a pure state (extremepoint of the set of states) of B + λJ. The procedure of [15] yields anirreducible representation φ of S for which z = kernel φ is the set{/: fe ίϊ, ρ(g*fh) = 0 for all g, h in β}. If p y e Range R(F3) then 9>(Λ, ) Φ0 and so zeί7,-. If p3- 1 Range i 2 ( ^ ) then φ(^(Ej)) = 0 and so zφE5.

In particular ze F1 = E and z$E2 = F. We have proved ze D but z0Zλ,for any j . This is a contradiction and so R(D) = ΣΠ=i R{D3).

Let ί7 = UΓ=i A = U?=i #< be in ^ , where A and E, are in 2? andDid Dj = φ = EiΓ\ Es \ί iΦ j. Then A f l ^ e ^ and

Σ Λ(A) = Σ Λ(A Π E3) = Σ Λ(£?ί)ΐ=i i,i=i i=i

Thus J? can be extended to & by the definition Jβ(F) = ΣΓ=i#(A),and the same reasoning shows that ϋ! is countably additive on &. Foreach p and g in lQ(φ), the function E —* (R(E)p, q) is a measure on ^?and can be extended to a measure μpq on , f. If B is a Borel set thenthere is a unique operator R(B) such that (R(B)pf q) = μM(2?) for allp, g. i2(ΰ) is a projection and B-+ R(B) is a projection valued measure.If Ee^f then we have already observed that R{E) is in the centerof the weak closure of f(B). By finite sums and monotone limits thisis true if E is a Borel set

If S is separable and type I and if &(f) is separable then Theorem1.9 is essentially known and in this case presumably the range of R isall projections in the center of the weak closure of ^($). If St is nottype I the range of R might not be this large, and in fact might be{0,1} even when the weak closure of ψ(&) is not a factor and is oftype J.

R is regular in the sense that for any open U, R(U) is the supremumof the R{K), as K ranges over the compact Borel sets in U. To seethis, let p be in ξ) and let / = /* be in & and vanish off U. Thenψ(f)p can be approximated by ψ{g)p, where g = g* and g vanishes offU9 - {z: \\f(z) || > ε} g {z: \\f(z) || ^ ε} = K2. U2 is open [8, Lemma 4.2]and ψ(f)p can be approximated by R{Uz)p and so by R(Ks)p. Ks iscompact [8, Lemma 4.3] and is a Borel set since Kz = Πo<δ<ε ϋi-

Proo/ of Theorem 1.6. Let φ, P be given as in the statement of1.6, let φ0 and ψλ be defined by Theorem 1.5 and 1.8 respectively, andlet R be defined by Theorem 1.9 in the case ψ = φx. If yeG, fe C0(Y),g e C0(X x G) and p e $(φ) then

since

/ ( ( , 7

-% σ)g(x,= ί

900 JAMES GLIMM

and since φ(j)φo(g) = φQ{g{Ί~λ', T"1-))- (See the proof of Theorem 1.5.)Let Ry be the projection valued measure defined on Z by Theorem 1.9in the case ψ = φ1ojκ. If U is an open subset of Z then

Me Πxez~u

)e Π λ~ = {<Pi{fM<P):fz Πxez~u J I xez~u

{ Π

and

Both E —> φ(j)R(E)φ(j~1) and E—> R(ΊE) are projection valued measureswhich we have just shown to agree with Ry on open sets. By theuniqueness part of Theorem 1.9, they both are equal to Ry and thus toeach other. This proves that φ, R is a representation of G, Z.

To show that R extends P, it is enough to show this for closedsubsets Έ of X. The range of J — P(E) is the closure of the set of

vectors \ f(x)dP(x)p where pe$(φ), feC0(X) and f(E) = 0. This

closure is also the closure of the vectors φ±{fA)p where A e S and/andp as before. To see this, use formula (1.10) and choose a suitable ap-proximate identity for S in C0(Y). The element fA of has the prop-erty (fA)(z) = 0 for z in π~\E). Let B be a self adjoint element ofί? and suppose £>(2) = 0 for z in π~\E). Let ε be a positive number.Then the set K = {z:\\ B(z) || ^ ε} is a compact subset of Z — π~\E) andτr(i£) is a compact subset of X disjoint from E. If g is a functionwhich is one on π(K) and zero on E then \\gB — B\\ < ε provided 0 ^g ^ 1. Thus the range of I — P(E) is the closure of the vectors φ1{B)pwhere pe £>(<£>), 5 e S and B(z) = 0 for 2 in n~\E). This is the rangeof I - R(π-\E)) so ^ ( T Γ - 1 ^ ) ) = P(#) and i2 extends P.

2 Induced representations. It follows from Mackey's work [11]that certain representations of G, X can be constructed in an explicitfashion from the action of G on X; these representations are calledinduced representations. In this section we determine the topologicalstructure of the space of all irreducible induced representations. This

space is homeomorphic to the orbit space $/G. Thus there is a corre-spondence between properties of Sί/G and properties of the induced repre-sentations; a simple example of this is Theorem 2.2.

Each φ in U determines a z in Z, namely z = k e r n e l φ e Z and thisz determines an x = π(z) in X. π(^) is the unique element of X suchthat all / in C0(Y) which vanish on {x} x Gxcz Y are in z. For any /in C0(Y), φ(f) thus depends only on values of / at {x} x Gx and ψdefines an irreducible representation φ1 of Lλ(Gx) and so of Gx. lί ψ isan irreducible representation of LX{GX) for some x in X, then /—•>'ίK/IM x GJ, / in C0(y), defines an irreducible representation ψ oΐ &,π(kernel ψ) = cc and ψ ~ ψ1. The map φ -+ φ1 preserves unitary equi-valence and so Jΐ is in one-to-one correspondence with the pairs x in Xand φ1 in Gx. The point x determines a correspondence between G/Gx,the right G* cosets, and the orbit Gx; Gxy corresponds to Ί~λx. Thiscorrespondence is a Borel isomorphism since the map GXΊ —* T"1^ is one-to-one and continuous and since the restriction of this map to a compactset is a homeomorphism. The induced representation Uφ\ Pφl, whichis a representation of G and G/Gx (G is transformation group acting onGIGX), defines by means of the correspondence Gxy«— j~xx a represen-tation Uφ, Pφ of G, X. By means of Theorem 1.5, Uφ, Pφ define arepresentation which we shall call Φ of C0(X x G) and so of S. If φ1

is irreducible, so is the joint action of Uφ, Pφ [11, §6] and so is Φ byTheorem 1.5. The map φ1 —* Uφ, Pφ preserves unitary equivalence [11,Theorem 2] as does the map Uφ, Pφ —> Φ (Theorem 1.5). Thus the map

Ψ —• Φ is a well defined map of & into 8. We recall that G acts on ί£by the map (7, cp) -^ cp T^1.

THEOREM 2.1. / / ^ and ψ are in S £/kβ Φ = ¥ if and only if φand ψ lie in the same orbit under G, that is if and only if there isa 7 in G such that ψ — φoyκ. The map φ—>Φ is continuous and theinduced map of the orbit space StjG is a homeomorphism with its image.

Proof. A. ψ — φ°yκ. Let φeSt and let x = τr(kernelφ). TheHubert space ξ>( Uφ) is the set of measurable functions / from G to £>(<p)such that f{σβ) — φ\o)f{β) for σ in Gx and β in G and such that the

integral \ 11/(7) \\2dμ(Gxi) is finite, where μ is some finite measure onJθlGx

G\GX which is quasi invariant. If ψ — φ°Ίκ then an / in C0(Y) is inkernel ψ if Ίκ{f) vanishes on {x} x Gx, which occurs if / vanishes on{j^x} x Gy~ix. Thus π(kernel ψ) = Ί~λx. Let v be the measure definedon G\GΊ-\X by means of the formula

where h e C0(G/Gγ-ix). This makes sense since Ί~λGxβ = Gy-i^β is aO7-ix coset, and one can see that v is quasi invariant.

If fe &(Uφ), let (Uf)(β) = /(7/5). Then C/jΓ is a measurable function

902 JAMES GLIMM

from G to $(φ) = &(ψ). If σeGy-ix then yσj~1eGx and (Uf)(σβ) =ffrσβ) = ^(ytrr1)/^) = ^(Ύσr-^Ufyβ) = f\σ){Uf)ψ). The last equ-ality follows from the fact that for gr in C0(F) and ?> in

, σ"1 ))ί> = ψ{c{

If f1e$(Uφ) also then

(2.1) ί ((ϊ7/)08)>(^/1)(/9))dy(Gϊ-i.iS)=( (f(β),Mβ)dμ(Gx

and since the right member of (2.1) is the inner product in ξ)(Uφ) andthe left member is the inner product in £>(t/^), Ufe&iU*) and U is aunitary transformation of &(Uφ) onto §(C7^).

Let £7 be a Borel subset of X. Then P ^ ) (resp. P ^ ) ) is multi-plication by the characteristic function of {β: β~λx e E) (resp. {β:β-'y-'x e E}) and

- U(P*(E)f)(β) ,

where χE is the characteristic function of E. Let a be in G. Thedefinition of Uφ(a)f = Uφ(ά)f is

1'*U*(a)f(β) = f(βa)(X(Gxβ, a))

where λ( , a) is a Radon Nikodym derivative of the measure E—>μ(Ea)with respect to μ. Then λ(τ , tf) is a Radon Nikodym derivative of themeasure E —> v(Ea) with respect to v and

= /(7/3α)(λ(G,7/3, α))1/2 = (Uϋ%a)f)(β) .

Thus ϊ/*, P^ is equivalent to Z7 , P^ and so Φ is equivalent to Ψ.

B. Φ = Ψ. Let <£> and f be in S and suppose that Φ is unitarilyequivalent to Ψ. Let # = π(kernelφ) and let y — 7r(kernel ψ). Pφ(Gx)is multiplication by the characteristic function of {β: β~λx e Gx} and soPφ(Gx) = I and likewise Pf(Gy) = I. (Gx is a Borel set since it is acountable union of compact sets.) Since Pφ and P^ are equivalent,P*(Gy) = I, Pφ(Gx Π Gy) = I,Gx Π Gy Φ φ and Gx = G?/. Suppose y =7x,yeG, and let ω = ψo^fκ. Then £? is equivalent to ?F by A, and sois equivalent to (P. Thus Uφ\ Pφl is equivalent to Uω\ Pωl and by [11,Theorem 2], ω1 is equivalent to φ1 and so ω is equivalent to >. Thusφ and n/r have the same orbits under G.

C. The continuity of φ—>Φ. The unitary equivalence class of the


induced representation is independent of the choice of the quasi-invariantmeasure μ on G/Gx. We make the choice μ = μx1 where μx is definedby the formula

(2.2) \ f(y)c(x, y^dy = \ \ f{σy)Ax{σ-')dσdμx{Gxy) ,Jσ JG/GXJGX

and fe C0(G). That (2.2) defines such a μx follows from Lemma 1.5 of[12] and its proof, and it is also shown there that Δ(y)c( "1x9 T)" 1 is aRadon Nikodym derivative of the translated measure E —> μx(Ey) withrespect to μx.

LEMMA. Let M be a compact symmetric subset of G and let s bea nonnegative element of CQ(G) which is positive on M. Then the

function t(x, 7) = s(y)[c(x, y)\ s(σ7)JJσ~1)dσ]-1 is defined and continuous

on the subset {(x, 7): Ί~λx e Mx} of X x G. If xeX and g is a boundedBorel function on G/Gx and if support g c GXM then

(2.3) ί g{Ί-λx)dμx{Gxy) = \ t(x, y)g(j-1x)dj .JθlQx JG

It is easy to see that t is defined and continuous. If g is continuousthen formula (2.3) follows from (2.2). The general case in which g isa bounded Borel function follows by taking monotone limits.

Let φm be a net of irreducible representations of β converging toan irreducible representation ψ. Let xm = π(kernel φm), let y—π(keτne\ ψ).If U is a neighborhood of y and if h is a function in C0(X) which iszero outside U and is one at y and if xm $ U then hSt c kernel φm. Theset {φ: h& ς£ kernel φ) is a neighborhood of ψ and so for large m,

h$t ςt kernel φm and xm e U. Thus xm —> y. The topology of ϋ can bedescribed in terms of w* convergence of linear functionals, and in par-ticular there are vectors vm in &(φm) and a w in tg(φ) such that || vm \\ =1 = \\w\\ and such that the linear functionals (φm( )vmf vm) converge inthe w* topology to (ψ( )w,w).

If fe C0(X x G), let f\i)(x9 σ) = f(x, σ~^). Then /°(τ) e C0(Γ) and7—•/°(7) is continuous in the norm || ||i and so in the norm || ||. Letφm' be the representation of GXm determined by φm. By [12, Lemma3.1], if

= φm(f°(Ύ))vm = \ f(xm, σ-1i)ψ

then F w €ξ)(Z7Oand likewise W = (7 -> f (f°(y))w) is in φ(!7*). Wesuppose that W ^ 0. This is the case for example if / is nonnegativeand has its support near X x e. If β and 7 are in G then

{{U«m{Ί)Vm){β), Vm(β)) - (V.,097), V . O S J M T M / S " 1 * , . , 7)-1])-1]1"

904 JAMES GLIMM

= (<Pm(Γ(β)* *f°(βv))vm, vm)[Δ{y)c{β-'xm, 7)-1]1'2

(2.4) - ( t ( / W *Γ(Pr)υ>, w)[Δ{Ί)c{β-ιy, y)~ψ2

= (W(βy), W{β))[Δ{Ί)c{β-ιy,Ί)~xT = {(U+(i)Wyβ), W<β))

and the convergence in (2.4) is uniform for β and γ in compact sets.Let g be in C0(X x G), let M be a compact symmetric subset of G

such that support f a X x M and let t(x, y) be chosen by the lemma.If βφGXm M then Vm(β) = 0 and we have

(Φm(g)Vm, Vn) =

= \ \ (sr(/5-X,7X[7*m(7)FJ(/3), Vm(β))dμXm(GxJ)dyJG jQ}Qχm

= \ \ t(xm,β)(g(β->xm,y)(U*m(y)Vm)(β), Vm{β))dβdy

t(y, βKgiβ-'y, y)(m(y)W)(β), W{β))dβdy

), W(β))dμy(Gυβ)dyJGlGy

= (Ψ(g)W,W).

This implies that Φm —> Ψ and proves C.

D. The induced map is a homeomorphism. It follows from what

we have proved that the map from Sϊ/G into 8 induced by the mapφ —>Φ is one-to-one and continuous. Let if be a closed G-invariant

subset of & and let L = {Φ; φ e K}. To complete the proof we must

show that L is relatively closed in the image of 5?.

Let f be in I , let Ψ be the corresponding element of 8, let^kernel^) = y, let g be in C0(Y), let h be in CQ(X x G) and let V andWbe in §(U+). Then

(W(g*h)W, V)

), V(β))dμy(Gyβ)dy

= \\ \ g(β-1y,σ)h(β-ιy,σ^y)((UHy)W)(β),V(β))JG JGlGy JGβ-ly

. [Jβ-ly(σ)IA(σ)γi2dσdμy(Gyβ)d7 .

The above integral is absolutely convergent and so we can interchangeorders of integration, placing the integration with respect to 7 first.If we substitute σy for 7, place the 7 integration last again, and thenuse the substitution σ-^β~ισβ as in (1.1), we obtain

(Ψ(g*h)W, V)

G JGlGy

= t tJθ J

(G/Gy JGy

=\\ \JG JGlGyJG

\ \JGlGyJGy

dσdμy(Gyβ)dΎ

= \ \JG j

), ψ o βκ(g*) V(β))dμy(Gyβ)dy .

Since the function β->φ<>βκ(g*)V(β) is in Φ(Ϊ7*),

(Ψ(g*h)W,V)=\ ((Ψ(h)W)(β), ψoβκ{g*)V(β))dμy(Gyβ)JGlGy

(ψoβκ(g)(Ψ(h)W)(β), V(β))dμυ(Gvβ) ,

and by limits converging in the norm in β, this is true for g in S.Let $ = {g; g e β and ^(g) = 0 for all 9> in #}. If ?F 6 L then

SΓC3;*8) = 0 by the above calculations. Now suppose Ψ is a limit pointof L. Then ξΓ(^*S) = 0 also. Since Ψ(2) contains a norm boundedsequence converging strongly to 7, if g e $ and F e §( t/^) then^ ° βΛΰ) V(β) = 0 for a.e./5. If we choose V continuous then β—>rf°βκ{Q)y{β) is continuous also; this can be seen directly if geC0(Y)and by taking uniform limits otherwise. For such V, ψ°βκ(9)V(β) = 0for all β. By [12, Lemma 3.2], this implies that ψ°βκ{g) = 0 and inparticular that i/r(S) — 0. By the definition of the hull-kernel topology,ψ e K~ = K,Ψ eL and L is relatively closed. This completes the proofof Theorem 2.1.

If x G X let <£>* be the one-dimensional representation / —> I /(a?, σ)dσ,

/GCoίF). Then φx can be extended to $, 9>β6ffi, kernel <px e Z anda? —> kernel ^ is a homeomorphism of X with its image in Z. Thisimage is invariant under G and so X/G is countably separated (thereare G invariant Borel sets E19 E2, in X which separate points of X/G)if Z/G is. However one might be interested only in representations

induced from a subset K of fi or of Z, and it is possible that K/G iscountably separated when X is not.

THEOREM 2.2. Let K be a closed G-ίnvariant subset of $£ and let

L be the closure of its image in 8. Let $(i£)(resp. $(£)) be the set ofg in $(resp. 8) for which ψ(g) = 0 if ψe if (resp. L). Then the fol-lowing statements are equivalent:

(1) 8/3(L) is type I( 2) K\G is countably separated(3) &I$(K) is type I and every factor representation of 8 which

906 JAMES GLIMM

annihilates $(L) is induced.

For a C*-algebra to be type I means that the weak closure of theimage of each representation is type / in the sense of Murray and vonNeumann.

Suppose (3) is true and let Φf be a factor representation of 8/^(L).Then the corresponding representation Φ of 8 is induced from a repre-sentation ψ of $. By Theorem 1.5 the commutant 0(8)' of 0(8) is theintersection of the commutants of Pφ and Uφ and by [13, Theorem 6.6],this is isomorphic to <p(^)' Since &ffi(K) is type I, φ(&)' is type / andso is Φ'W(L))r. Thus Φf is type I and so is 8/3 (L), and (3) =φ (1).

Suppose (1) is true. By [5, Theorem 2], L is countably separated andby Theorem 2.1, KjG is homeomorphic to a subspace of L. Thus K\G iscountably separated, and (1) => (2).

Suppose (2) is true. If xeX, let K(x) be the set of φ in K suchthat π(kernelφ) = x. If jeG and ψ and φ°Ίκ are both in K(x) thenjeGx and φ is equivalent to φ°Ύκ. Thus the restriction to K(x) ofthe quotient map K-+K\G is one-to-one. Let JEΊ, E2, be G invariantBorel subsets of K which separate the points in K\G and let Ul9 U2,be open subsets of X which separate points of X. Then ^(U^^CU^,—separate points of K (x) from points of K(y) for xΦy and El9 E2y —separate points of K(x). Thus K is countably separated and by [5,,Theorem 2], RI3(K) is type /.

Let φQ be an irreducible representation of 8 which annihilates 3(L),let φ and P be the corresponding representations of G and X and letR be the projection valued measure on Z which extends X and is givenby Theorem 1.6. We assert that R(Z ~ K) — 0. Let ψx be the repre-sentation of S defined by Theorem 1.8. In view of the definition of R,we must show that ^^(K)) = 0. Suppose first that φ0 = Ψ is inducedfrom an irreducible representation ψ of ££ which annihilates 3( iO andlet # be in $(K) and T7 in ξ>(0>). As in the proof of Theorem 2.1,A (ti(ff) W)(β) = f o βκ(g) W{β) for a.e. β, and so ψ^g) = 0 and ψ^iK)) =0. If we no longer assume that <p0 is induced, φ0 is in any case a limitof such induced representations Ψ. Thus if W and Feξ>(<p0) andΛ, e C0(X x G) the representative function

g-+(Ψi(g)<Po{h)W, V) = (φo(g*h)W, V)

defined on Co( Y) is a limit of uniformly bounded representative functionsdefined on 5Ϊ and vanishing on $(K). This implies that t i (3( ίQ) = 0and R(Z ~ K) = 0.

Since the images of


ψ0 are induced. This means that the map of K\G —* L is onto, that Lis countably separated and by [5 Theorem 2] that 2ffi(L) is type I.We have proved that any irreducible representation of 8 which annihilates$(L) is induced and thus this is also true for factor representations.We have proved (2) =Φ> (3), and this completes the proof of Theorem 2.2.

Some of the results of this section extend results of [3], and thispaper is in part addressed to the problems considered in [3] (cf. Thefinal paragraph of [3]).

We conclude with a proof of the result mentioned in the introductionconcerning a manifold structure in orbit spaces. We are indebted toR. Palais for discussions concerning this theorem.

THEOREM 2.3. Let Kbe a C°° or real analytic separable n-dimension-al manifold and let G be an analytic group acting smoothly on K. Ifthe orbit space K\G is countably separated and if the orbits all havedimension m then there is an open dense G invariant subset U of Kand a unique C°° or real analytic n-m dimensional manifold structureon UjG such that a function f defined on UjG is differentiate ( = C°°or real analytic) near Gx if and only if the corresponding functionx-+f(Gx) defined on U is differentiate near x.

If K\G is countably separated then Theorem 1 of [6] implies thatthere is a dense open G invariant subset Uλ of K such that UJG is T2;we can suppose K = Ux. If x e K, let θx(j) — yx, for 7 in G. If Γ e g,the Lie algebra of G, let Θ+(Γ) be the vector field defined by Θ+(Γ)X =dθx(Γ). Then Θ+(Q) is an m-dimensional involutive differential systemUJΪ on K, by [14, page 35, Theorem 2]. Necessary and sufficient conditionsfor coordinate functions xlf •••,£„ to be flat with respect to SQΐ(we usethe terminology of [14]) is that Xj(yy) = Xj(y) for y near e, y in thedomain of the xk and j = m + 1, , n. Suppose this is the case, supposethat the coordinate system is cubical of breadth 2a and domain Wa andlet S = S(cm+1, , cn) denote the slice {x; xά{x) = cjf j = m + 1, , n}of Wa. Let x be in S. Since dθx maps g onto 9JΪX, θx maps each neighbor-hood of e onto a neighborhood of x in S. Let T be the leaf containing S.Since each y in T is in some such S, T Π Gx is an open subset of T in themanifold topology for T as a submanifold of K. Since K\G is Γ2, Gx isclosed and T Π Gx is a relatively closed subset of T with the relativetopology and so is a closed subset of T in the manifold topology. SinceT is connected in the manifold topology, TaGx. For some neighborhoodN of e, Nx c S, and then {7; Ύ% e T} can be shown to be an open andclosed subset of G and thus all of G. Thus the leaves are the orbits.

Let W be a G invariant open subset of K. We show that W con-tains a G invariant open subset consisting of regular leaves. This willcomplete the proof since the union U of all open G invariant subsets

908 JAMES GLIMM

of K which consist of regular leaves will then be dense, and [14, Theo-rem 8, page 19] defines the required manifold on U/G. Let Wz ={x: \Xi(x) I < ε}. There is an ε in (0, a) and a neighborhood N of e suchthat

N(S(cm+1, , cn) Π Wζ) c S(cm+1, . . . , c , )

for all cm+u •••,<?». By Theorem 1 of [6] there is a nonempty opensubset UQ of Wz such that for each m in UQ, Nm Π Uo = Gm Π Z70 IfS(cm+1, . . . , c . ) Π tfo=£0 then

(GS(cm+u , cn)) nuo = (G(S(cm+1,..., o n u0)) n u0

= (N(S(cm+1,..., cn) n I7O)) n c/o = S(cm+1, , o n t/Όand so each orbit that meets Uo meets it in a set of the formS(cm+U , cn) Π UQ. It follows that each orbit through Uo is a regularleaf and that GU0 is the required open subset of W.

D. Mumford has constructed an algebraic quotient using relatedhypotheses (Conversation with A. Mattuck).

APPENDIX

J. M. G. Fell has proved the equivalence stated on the first page ofthis paper. What follows is his proof.

Let G be a locally compact group with unit e and let £f be thefamily of all closed subgroups of G. Let us give to & the topologyhaving as a basis for its open sets the family of all

: K n C = φ,K Π AΦ φ for each A in

(where C runs over the compact subsets of G and ^ runs over thefinite families of nonvoid open subsets of G). This topology makes S^a compact Hausdorff space [4, Theorem 1], Let us fix a nonnegativefunction f0 in C0(G) such that fo(e) > 0 and for each K in £f let μ κ bethe left Haar measure on K for which

*) = l .

THEOREM. .Fbr βαc/ / in C0(G), the function

is continuous on £f.

First, we observe that to each compact subset C of G there is apositive number a = a(C) such that

( 1) μΣ(C f] K) ^a

for all K in S^ In fact if fo(z) > ε > 0 for all z in a neighborhood Uof e and if x e C then choose a neighborhood £7 of # such that J7ίB~

1Z7a.c C/.A finite number of these, £7 , , UXn, cover C. Let a = w/ε, let J ={i; UXJ Π K Φ φ) and if j" e J, let y3- be chosen in C/ Π if. Then

/^(C fl if) ε~3

The essential technique is that of generalized limits. Let Kn be anet in & converging to if and let Kn be directed by a set N. Ageneralized limit is a positive linear functional Γ defined on the spaceB of all bounded real valued functions on N such that if s e B andlimTC_>oo sn exists then Γ(s) = limΛ_oo sn. If s e B and Γ(s) is the same forall possible generalized limits, then lim^βo sn must exist and equal Γ(s).

Now let Γ be any generalized limit and let / be in C0(G). By (1),

the function \ f(k)dμκ (k) defined on N is bounded. Let

f{k)dμEβc)n

Φ is a positive linear functional on C0(G). If / = 0 on if, choose f5 inC0(G) converging to / uniformly and such that the support of /δ iscontained in {x: \f(x)\ ^ δ}. Then ^(suppt /δ, φ) is a neighborhood ofif and if Kn is in this neighborhood then I fs(k)dμKn(k) = 0 and so

φ(fs) = o and Φ(f) = 0. Also every g in C0(if) extends to a n / i n C0(G),so the definition

φ(f\K) = Φ(f), feC0(G)

gives a positive linear functional ψ on C0(K).If koe K and if ε > 0 then by (1) we can choose an open neighbor-

hood U of kQ such that

r

< ε

for all kx in U and i ϊ in &. For large %, iΓTC e <&($, U) and so thereis a &„ in Kn Π Z7. Hence

^ lim supΓ f{Kk)dμKn{k) - f{kk)dμKn(k)

+ lim sup

^ ε | | Γ | | + lim supΓ(\ f(k)dμKn(k))-φ(f\k)\j κn /

= e | | Γ | | ,

910 JAMES GLIMM

so φ is left invariant on K and thus is a left Haar measure. Since

φ(f01 K) =

we must have

Φ{f) = ( f(k)dμκ(k)

for all / in C0(G). The right member of the previous equation is inde-pendent of the choice of Γ and hence so is the left member. Thus

lim ί f(k)dμKn(k) = ( f(k)dμκ(k) ,n Jκn JK

and the theorem is proved.If Gx is a continuous function of x and if μx = μGx is chosen as

above then x —> μ% is a continuous choice of the Haar measures. Con-versely suppose we are given a continuous choice x —> μx of Haar measureson the Gx and suppose that {xn:neN} is a net in X converging to yand that ^/{K, &~) is a neighborhood of Gy. If GXw Π K is not eventu-ally empty then for all n in a coίinal subset of AT, there is a σn inG^ Π if, and if we pass to a suitable subnet, σn—»σ. However σ e K ΠGy

which contradicts the fact that %/(K, ~) is a neighborhood of Gy. LetVe Jt~ and let / be a nonnegative nonzero element of C0(G) with supportin V. Then ί f(σ)dy(σ) > 0 and so ί f(σ)dx (σ) is eventually greater

that zero. Hence GXγι Π F is eventually not empty, GXfl is eventuallyin %s(K, ~), and Gx is a continuous function of cc.

REFERENCES

1. R. Blattner, Positive definite measures, Submitted to Proc. Amer. Math. Soc.2. J. Dixmier, Algebres quasi-unitaires, Comment. Math. Helv., 26 (1952), 275-321.3. J. M. G. Fell, Weak containment and induced representations of groups, CanadianJ. Math., 14 (1962), 237-268.4. , A Hausdorff topology for the closed subsets of a locally compact non Hausdorffspace, Proc. Amer. Math. Soc, 13 (1962), 472-476.5. J. Glimm, Typel C*-algebras, Ann. of Math., 7 3 (1961), 572-612.6. , Locally compact transformation groups, Trans. Amer. Math. Soc, 101 (1961),124-138.7. P. Halmos, Measure theory, Van Nostrand, New York, 1950.8. I. Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc,7O (1951), 219-255.9. L. Loomis, An introduction to abstract harmonic analysis, Van Nostrand, New York1953.10. , Positive definite functions and induced representations, Duke Math. J., 27(1960), 569-580.11. G. Mackey, Imprimitivity for representations of locally compact groups, Proc Nat.Acad. Sci., U. S. A. 35 (1949), 537-545.


12. , Induced representations of locally compact groups I, Ann. of Math., 55 (1952),101-139.13. , Unitary representations of group extensions I, Acta Math., 99 (1958), 265-311.14. R. Palais, A global formulation of the Lie theory of transformation groups, MemoirsAmer. Math. Soc. no. 22 (1957).15. I. Segal, Irreducible representations of operator algebras, Bull. Amer. Math. Soc , 53(1947), 73-88.16. , Two-sided ideals in operator algebras, Ann of Math., 50 (1949), 856-865.

ON ALMOST-COMMUTING PERMUTATIONS

DANIEL GORENSTEIN, REUBEN SANDLER AND W. H. MILLS

Suppose A and B are two permutations on a finite set X whichcommute on almost all of the points of X. Under what circumstancescan we conclude that B is approximately equal to a permutation whichactually commutes with At The answer to this question depends stronglyupon the order of the centralizer, C{A), of A in the symmetric groupon X; and this varies greatly according to the cycle structure of A,being comparatively small when A is either a product of few disjointcycles or a product or a large number of disjoint cycles of differentlengths and being comparatively large when A is a product of manydisjoint cycles, all of the same length. We shall show by examplethat when the order of C(A) is small there may exist a permutation Bwhich commutes with A ' 'almost everywhere" yet is not approximatedby any element of C{A). On the other hand, when A is a product ofmany disjoint cycles of the same length, we shall see that for any suchpermutation B, there must exist a permutation in C(A) which agreesclosely with B.

It is clear that if B is a permutation leaving fixed almost all pointsof X, then no matter what permutation A is given, B will commutewith A on almost all points of X, and at the same time B can beclosely approximated by an element of C(A)—namely, the identity.However, the examples we shall give will show that only when all (ornearly all) of the cycles of A are of the same length can we hope toapproximate every B which nearly commutes with A by an element inC{A). Accordingly, the bulk of this paper will be taken up with thestudy of the case in which A is a product of many disjoint cycles, allof the same length.

1* In order to get a satisfactory notation and a more compact wayof discussing the problem, we begin by making the symmetric groupSN(X) on the space X into a metric space. Here N denotes the cardi-nality of X, and it is to be understood that N is finite. Define, forany A in SN(x),

(i) IIA| | = _ Λ

where fA is the number of fixed points of A on X Now define thedistance d(A, B) between two elements A and B of SN(X) to be

(2) d(A,B)=\\AB-1\\.

Received December 2, 1961,

913

914 DANIEL GORENSTEIN, REUBEN SANDLER AND W. H. MILLS

Under these definitions, the identity is the only permutation of norm0, every permutation has norm ^ 1, and a permutation has norm p ifand only if it moves pN points of X. In particular, the permutationsA and B commute if and only if || [A, B] || = 0, or equivalently, if andonly if d(AB, BA) = 0.

In order to see that these definitions make SN(X) into a metricspace, we need only verify the triangle inequality, since the other prop-erties are trivial. But the points of X displaced by AB are clearlyamong those which are displaced by either A or B. Hence N — fΛB ^(N - fA) + {N- fB) and consequently || AB || ^ || A || + || B ||. We thushave the following lemma.

LEMMA 1. With the norm defined above, SN(X) forms a metricspace.

When no restriction is placed upon the cycle structure of A, wehave the following result:

PROPOSITION 1. For any ε > 0, there exists an integer N andpermutations A and B in SN(X) such that || [A, B] || < ε and such thatd(B, D) = l for every D in C(A).

Proof. We shall give two examples of permutations A and B whichsatisfy the conditions of the proposition; in the first, A will be a productof cycles of relatively prime lengths, and in the second, a product ofcycles of lengths n and 2n.

EXAMPLE 1. Let X = {1, 2, , N}, where N = 2n > 4/ε. Let Abe the permutation

(1 2 n - l)(n)(n + 1 n + 2 2n)

and B the permutation xB — x + n if x ^ n, and xB = x — n if x > n.By direct verification, we find that A and B commute except on thepoints n — l,n,2n — 1, 2n. Thus / U ) β ] = N — 4 and hence || [A, B] \\

On the other hand, any element D of C(A) must map each cycleof A into itself, since these cycles are of different lengths. But, forany x in X, xB and x lie in distinct cycles of A. It follows that forany D in C(A), BD~X displaces every point of X and hence thatd(B, D) = 1.

EXAMPLE 2. Let X= {1, 2, •••, N}, where N= Anm and n > 1/e.Let A be the permutation with m cycles of length 2n and 2m cycles

ON ALMOST-COMMUTING PERMUTATIONS 915

of length n, defined as follows:

(1 2 2n)(2n + 1 4n) (2n(m - 1) + 1 2nm)

(2nm + 1 2nm + n)(2nm + n + 1 2nm + 2n)

(Anm — n + 1 Anm) .

Let B be the permutation xB — x + 2nm if x ^ 2wm, and xB =α? — 2nm if x > 2nm.

Again, by direct computation, we find that A and B commute onall points x of X except when x = 0 (mod n). Thus fίA,Bi ~ 4wm — 4mand hence \\[A, B]\\ = 1/n < e. On the other hand, if DeC(A), Dmust permute the cycles of A of length n among themselves and mustpermute the cycles of A of length 2n among themselves. But if x isin a cycle of length n9 then xB is in a cycle of length 2n, and viceversa. It follows that BD~ι displaces every point of X and hence thatd(B, D) = 1, for any D in C(A).

2. The two examples given in Proposition 1 indicate that unlesssevere restrictions are placed on the cycle structure of A, the fact thatB comes very close to commuting with A does not necessarily implythat B can be approximated by an element in C(A). In fact, it seemsthat unless A consists almost entirely of cycles of the same length, littlecan be said in general of the relation between || [A, B] || and the dis-tance from B to C(A).

In order to be able to make as exact statements as possible, weshall assume in the balance of the paper that A is the product of mdisjoint cycles, each of length n. In this case our statements aboutthe distance from B to C(A) will depend only upon || [A, B] \\ and n.

We may take X = {1, 2, , N}, where now N = nm. Let x, k beintegers such that 1 S % ^ N9 0 ^ k ^ n, and write x = in + r, where1 ^ r ^ n. We shall adopt the following notation:

(3) x + k = in + s, where 1 ^ s ^ n and s = r + k (mod n) .

Without loss of generality we may assume that A is the mapping

(4) xA = x + 1, x e X .

We shall say that B in SN(X) transforms the cycle a of A intothe cycle α' if, for some x in α, #1? is in af and

\O) \tΛ/ ~T~ n/JJ-β tλ/JL> ~\~ ΓVy tv — \ ) , J-, > Iv X

We shall write (α)# = af it B transforms a into α\ We shall alsosay that B commutes with A on a cycle a if it commutes with A oneach point of a.

LEMMA 2. (a) A permutation B commutes with A on a cycle aif and only if B tr an forms a into a cycle a\

(b) if B commutes with A on n — 1 points of a cycle α, then Bcommutes with A on a.

(c) // B transforms r cycles of A into cycles of A, there existsan element D in C(A) which agrees with B on these r cycles.

Proof. For A and B to commute on a point x of X we must havexBA = xAB, and hence

(6) xB + 1 = (x + 1)B .

Suppose (a)B = a!\ then (6) follows at once from (5) for any x ina. Conversely if (6) holds for all x in α, (5) follows at once by induc-tion on k.

To prove (b), suppose B and A commute on x, x + 1, , x + n — 2.Again by induction on k, (5) holds for k = 0,1, , n — 2. In particular,(x + n — 2)B = xB + n — 2. Now using (6) with x replaced by x + n — 2,we obtain

(x + n - 1)5 ={x + n — 2)B + 1

= xB + n — 2 + 1 = xB + n - 1 .

Thus (5) holds for all k, and hence A and B commute on a by part (a).Finally suppose B transforms the cycles alf * ,α r into the cycles

a[, « ,α' . Denote by a'r+u •• ,a'm the remaining cycles of A. Let Dbe a permutation which agrees with B on al9 , ar and transforms a{

into α , i = r + 1, , m. By (a) D is in C(A).

3 We shall now begin the analysis of the relationship between\\[A, 5] || and the minimum distance from B to C{A), under the as-sumption that A is the product of ^-cycles. We shall denote thisminimum distance by dA(B). Thus

(7) dA(B) - min d(B, D) .D€C{A)

Then following estimate for dA(B) is easily obtained.

PROPOSITION 2. For any B in SN(X),

; nil [A, BUI

Proof. If || [A, B] || ^ 2fn, the proposition is vacuously true sincedA(B) ^ 1. Hence we may assume that || [A, B] || < 2\n.

Now N = nm, where m is the number of cycles in A. It sufficesto show that B transforms at least

m N\\\A,B]\\

cycles of A into cycles of A. For then by Lemma 2(c) we can find anelement D in C(A) which agrees with B on these cycles and hence onat least

N-ψ-.\\[A,B]\\

points of X. It follows that

, D) <L "IIIΛfllll .

By the definition of || [A, B] ||, N || [A, B] || is the number of points-displaced by [A, B] and hence on which A and B do not commute. Butby Lemma 2(b) any cycle of A which is not transformed by B into acycle of A contains at least 2 points on which A and B do not com-mute. Thus there are at most

N\\[A,B]\\

cycles of A which are not transformed by B into cycles of A, and henceB transforms at least

m N\\[A,B]\\2

cycles of A into cycles of A.Proposition 2 gives an upper bound for dA(B), which depends only

upon || [A, B] || (and ri), but not upon the particular structure of B.Our main concern in the paper will be in improving this upper bound.The next proposition shows the limit to which this estimate can beimproved.

PROPOSITION 3. If A contains at least two distinct cycles, thenthere exists a permutation B in SN(X) such that

d A i B ) = " I I 1 ^ * 1 II

when n is even, and such that

918 DANIEL GORENSTEIN REUBEN SANDLER, AND W. H. MILLS

when n is odd. Furthermore for any ε > 0, N and B can be chosenso that || [A, B] \\ < ε.

Proof. Assume first that n is even. Set m — m1 + m2, wherem1 Ξ> 0 and m2 ^ 2. Define the permutation B as follows: xB = x if1 <L x < nrnj; if x > wm^ write x = in + k where 1 < k ^ n, and define#j? = x if & ίg w/2, xB = x + n if i Φ m — 1 and & > w/2, and xB =ίM»! + fcifΐ = m — 1 and fc > w/2.

Thus 5 leaves the first m1 cycles of A pointwise fixed, one half ofeach of the remaining m2 cycles pointwise fixed, and permutes the otherhalves of these ra2 cycles cyclically. From its definition, we see thatB commutes with A except on the points x > nm1 for which x = 0(mod n\2). Thus

(8) \\[A,B}\\= ~N

Since N = n{m1 + m2), 2mJN can be made arbitrarily small bymaking m1 sufficiently large. Thus, to prove the proposition, we haveonly to show that

Observe, first of all, that the identity, /, is in C(A) and agreeswith B on

tm. + Jϋϋb-

points of X, whence

(9) d(I, B) = S — = Jϊ!22- = ΊL || [A, B]

On the other hand, by Lemma 2, any D in C(A) must transformeach cycle a{ of A into some other cycle α, . Since B transforms thetwo halves of the cycles α< into distinct cycles of A, m1 ^ i ^ — 1, Z>and 5 can agree on at most half of the nm2 points in these cycles.Hence DB'1 displaces at least nm2/2 points of X, which implies that

= £\\[A,B]\\

for any D in C(A).When n is odd, the construction of B is entirely analogous.


4; If we set

dA = max A\• * — ,BesNiχ) \A, B\\\n

then dA is a measure of the extent to which every permutation inSN(X) can be approximated by elements in C{A). Propositions 2 and 3show that

according as n is even or odd.In the balance of the paper we shall sharpen these inequalities by-

lowering the upper bound for dA. Our next result will show that inconsidering this problem, we may restrict our attention to those cyclesof A on which B commutes with A on exactly n, n — 2, or n — 3 points.Let UB, VB, WB be the set of points in those cycles of A on which Bcommutes with A on n, n — 2, and n — 3 points respectively; and letuB=\UB\,vB=\ VB \,wB=\ WB |.

THEOREM 1. Suppose there exists an element D in C(A) whichagrees with B on at least uB + {l\2)vB + (ljS)wB points of X. Then

Proof. For simplicity of notation, we drop the subscript B, anddefine

(11) t = N— u — v — w .

Thus t is the number of points in those cycles of A on which A andB commute on no more than n — 4 points. Then by definition of u,vf

w, t, we have

(12) u ^ Ά Ln n n

Now, by hypothesis,

N- (u + —v + —w\ —v + —w + t(13) d(Bf D) £ ^ ^ ^ — = -? ^

We must show that

(14)


But using (1), we can rewrite (14) as:

(15) fUM ^u + 2^lv + (i _ 8 V,

Since (15) is an immediate consequence of (12), the theorem follows.

5 In this section, we prove that dA ^ 1/4, by proving that forany B in SN(X), there exists a permutation D in C(A) which satisfiesthe conditions of Theorem 1.

To treat our problem, we need an additional concept: By a block

of a cycle a of A, we shall mean a maximal sequence x, x + 1, •• ,

x + r — 1 of points of a such that A and B commute on every point

of the sequence except x + r — 1. The integer r will denote the length

of the block. According to the definition, if A and B commute on

every point of a then a contains no blocks. When B and A do not

commute on every point of α, we have the following obvious lemma:

LEMMA 3. If A and B commute on exactly n — k points of acycle a of A, k > 0, then A contains exactly k blocks, the sum of whoselengths is n.

Thus when a cycle a of A lies in VB, a consists of 2 blocks whichwe denote by p19 p2; and when a lies in WB9 a consists of 3 blocks whichwe denote by qu q2, q5. We define \p3 \, \ q^ \ to be the lengths of pj9 qjf

respectively. Furthermore we order the blocks so that | p± \ | p21 andI q11 ^ \q21 ^ I 9.1- Since | px \ + | p2 \ = n,

(16) Iftl^f

and likewise

(17) kil^x

Let x9 x + 1, , x + r — 1 be a block contained in a cycle a. IfxB = -?/, then, it follows from (6) as in the proof of Lemma 2, that

(18) (x + k)B = y + Λ , 0 ^ & r - 1

and

(19) (x + r)B Φ y + r .

Thus the image of the block is a consecutive sequence of points in acycle af. It follows that there exist permutations which transform a


into a1 and agree with B on the block 6 = {x, x + 1, , x + r — 1}.In fact, any D in C(A) for which xD = y has this property. If D issuch a permutation, we shall write simply (a)D = α'; (δ)Z> = (δ)i?.

From this fact, we easily derive the following lemma:

LEMMA 4. Let a19 •• ,αfc be distinct cycles of A containing theblocks blf , δfe respectively. If the images of 6; under B lie in dis-tinct cycles a\ of A, i = 1,2, •• , k, then there exist permutations Din C(A) such that (a^D = a[; {b%)D = (b^B, i = 1, 2, , k.

We are now in a position to prove the following result:

THEOREM 2. Given any B in SN{X), there exists an element D inC(A) which agrees with B on at least

points of X.

Proof. Let alf a2, , am be the cycles of A. For any i, j, 1 ^ i,i ^ m, let δ^ be the maximal number of elements of a{ on which apermutation D in C(A) mapping a{ into c^ can agree with B. Thus ifi? transforms a{ into α,, δ^ = n. If (α^β Π aό — ψ, then δί:/ = 0. Now,to any mxm permutation matrix {eia) there corresponds a permutationD in C(A) which agrees with B on

(20) Σ ^ Ap-

points, where D is defined to transform α, into aj if e{j = 1, and to

map α so as to agree with B on δ o points.We wish to show

(21) max Σ eφa ^ u + \v + ^w ,

where {ei3) ranges over all permutation matrices. To do this, considerthe set of all real mxm matrices {xi5) such that

(22) xi:i ^ 0 1 ^ i, j ^ m

(24) Σ * w = 1 l^ί^m.

This is the set of doubly stochastic matrices and is a convex, boundedset whose vertices consist of exactly the permutation matrices (see [1],

922 DANIEL GORENSTEIN REUBEN SANDLER, AND W. H. MILLS

pp. 132-3).The following lemma will be useful in proving the theorem.

LEMMA 5. If (xi3) is any doubly stochastic matrix, then thereexists a permutation matrix (eί3) such that

(25) Σ ei3bi3 ^ Σ,xi3bi3 .i.3 i.3

Proof. See [1], p. 134.If we can now demonstrate a doubly stochastic matrix such that

(26) 5X A, ^u + h) + —wf

we will clearly be finished since, by Lemma 5, there must then be somepermutation matrix (ei3) such that

Σ eiόbi3 ^ u + —v + —w ,

and this permutation matrix will yield the desired mapping D.To find a matrix satisfying (26), define

(27) xi3 = ^ ~ ,n

where nl3 is the number of points of a{ which B maps into a3. Thematrix (xi3) is clearly doubly stochastic, so we must show that (26)holds. But if a{ gΞ UB, then

since (a^B — aj± for some j \ . If a{ S Vβ, there exist indices j \ and j 2

such that (p^B c α i χ and (p2)B c α i 2. Note that j \ Φ j 2 , or else α*would be transformed by B into α^. In this case, then,

(remember | Pi | + | pa I = ^)

Finally, when a{ s Wΰ, one of three things can happen:(a) ?i, ?a> 3 can be mapped by i? into three distinct cycles of A.(b) qlf q2, q3 can be mapped by B into only two cycles of A,(c) ?i, ^2, 3 can be mapped into one cycle of A.

In the first case,


In the second case,

where | qH | ^ | qH \ .

Finally, in case c,

where | gx | ^ | q21, | ? 8 1 .

Since | qx | + | q2 \ + | g31 = n, it follows at once in all three casesthat

3 O

We have thus demonstrated the existence of a doubly stochastic matrix(Xij) with the property

Σι%isK' ^ u + TΓ v + i r w

ί.i 2 3

Together with Lemma 5, this completes the proof of the theorem.As an immediate corollary of Theorems 1 and 2, we obtain our

main result:

THEOREM 3. Let A contain at least two distinct cycles. If n iseven, dΛ = 1/4. If n is odd,

— ^ dA ^ — .4^ 4

REFERENCE

1. S. Karlin, Mathematical Methods and Theory in Games, Programming and Economics,vol. I (1959).

CLARK UNIVERSITY

INSTITUTE FOR DEFENSE ANALYSES

YALE UNIVERSITY

CONGRUENCE PROPERTIES OF σr{N)

V. C. HARRIS AND M. V. SUBBA RAO

l Introduction. Let σr(N) denote as usual the sum of the rthpowers of the divisors of N. Let d be a divisor of N with 1 ^ d <£ λ/Nand df its conjugate, so that ddf = N. By a component of σr(N) wemean the quantity dr + drr or dr according as 1 ^ d < τ/JV or cί = τ/JV.Components corresponding to distinct divisors d :g "l/iV are distinct andσr(N) is their sum.

If every component of σr(N) is congruent to the integer α, moduloK, we say that σr(N) is componently congruent to a (mod if) and indi-cate this by writing

σr(N)=a (mod K) .

This does not necessarily imply that also σr(N) = a (mod if). For example0*4(8) = 2 (mod 3) but <74(8) = 1 (mod 3). Similarly ordinary congruence doesnot imply component congruence, as the same example shows.

2. THEOREM 1. If r, if, L are fixed positive integers with K Ξ> 3and (L, if) — 1, and if a is a nonnegative integer, then a necessaryand sufficient condition that

(1) σr(nK + L) = a (mod if) for all integral values of n^ 0

is that

(2) L is a quadratic nonresidue of K

(3) 1 + Lr = a (mod K)

(4) (wr - l)(wr + 1 - α) = 0 (mod K) for all w such that (w, K) = 1

We first show necessity. Assume that σr(nK + L)=a (mod K) andL is a quadratic residue of i£. Then there exists q such that q2 = L (mod iί)and consequently nx such that ^i f + L = q2. Consider q2 and n2K +L = {nλK + nλ + L)K + L — (K + l)tf2, both occurring in the sequencenK + L. Since σr(q2) = a (mod iί) we have with d = q that gr Ξ a (mod if)and since σr([K + Ϊ\q2) = a (mod K) we have with d = q and df = (K + l)qthat qr + (if + l)rqr = α(mod iΓ). Thus qr + (K+l)rqr = q\ or, 2 = l(modiΓ).This is a contradiction and (2) is necessary. Assume next (1) holds.Then in particular for n = 0 we have σr(L) = a (mod K). By condition(2) just proved L Φ 1 and the component with d = 1 and df = L gives1 + Lr=a (mod if) which is (3).

Received May 11, 1961, and in revised form November 1, 1961,

925

926 V. C. HARRIS AND M. V SUBBA RAO

Next to show (4). Given any w such that {w, K) — 1, there existsan x ί W (mod K) such that wx = L (mod K). Let this x be denoted by

Then

ww1 Ξ L (mod K)

and by our assumption σr(nK + L) = a (mod K) applied to ww1 it followsthat

1 + wrw{ = a (mod K)

wr + w\ = a (mod K) .

Eliminating w\ gives 1 + wr(a — wr) = α (mod iΓ). Rewriting this gives(4) and shows (4) is necessary.

To show sufficiency, we need to show for any divisor d of N =nK + L with 1 ^ d ^ i/iNΓ and conjugate divisor d' thatcίr + dtr =a (mod if) or dr = a (mod ίΓ) according as 1 ^ cί < VNOY d = VN provided(2), (3) and (4) hold. But (2) insures that N cannot be a square, sothe second alternative cannot occur. Now

dr(dr + d») = d2r + (ddj

= (1 + adr — a) + Lr

by (4) and the fact that ddf = L (mod iΓ). Then using (3),

dr(dr + d'r) = (1 + adr - α) + a - 1 Ξ αώr (mod K).

Since (cZ, K) = 1 it follows that

for each d as specified. But this shows (1) holds and completes the proof.

3* Examples and some special cases. It is not difficult to show thatwhen K = p is an odd prime, all component congruences are obtainedwith r = (p — l)/2 and a = 0 or r = (p — 1) and a = 2. Thus for example:

σe(lSn + L) = 0 (mod 13), L = 2, 5, 6, 7, 8,11

<712(13π + L) = 2 (mod 13), L = 2, 5, 6, 7, 8,11 .

When K is composite we have σφ{κ){nK + L)==2(mod iΓ) for any non-quadratic residue L of K.

In the special case r = 1 we show

THEOREM 2. For αίZ integral n^O, σ^nK + L)=α(modiί)/or suitable L and a if and only if K is one of 3, 4, 6, 8,12 and 24.

The equation in condition (4) becomes

CONGRUENCE PROPERTIES OF σr(N) 927

( 5 ) w2 — aw + a -1 =0(modK)

The congruence (5) is equivalent to

Ax2 — Aax + a2 = (2x — a)2 = (a — 2)2 (mod 4K) .

With y — 2x — a we have

(6) y2 = (a-2)2(modAK)

subject to y= —a (mod 2). But this last condition is no restriction sothat the number of solutions of (5) is the same as that of (6). LetS(AK) be the number of solutions of (6) and let AK = pζ+βlp? p)' wherepx — 2, p2 = 3, are distinct primes. Then

S(AK) = S(pl+ei)S(pl*2) S(py) and S(pl+e^) ^ 2 for eλ = 0

+βl) ^ 4 for e > 0; S(p?) ^ 2 for ^ > 2 .Since (5) is to hold for all w such that (w, K) = 1, we must have

or

4 ^ = 2, βi > 0 .

2 Pι>2

(7) P'rKPi - i) = Φ(PΪ) <

But the only values of p\ι satisfying these are 1, 2, 4, 8 and 1, 3. SinceK ^ 3 these give K = 3, 4, 6, 8,12, 24. The converse can be proved byenumeration. The results are listed:

K 3 4 6 8 8 12 12 24 24

L 2 3 5 3 7 5 11 11 23

α 0 0 0 4 0 6 0 12 0

4» Relation between component congruence and congruence • Wehave

THEOREM 3. If σr(nK + L) = α(mod K) for all integral n ^ 0, thenσr(nK+ L) = a (mod K) for all integral n Ξ> 0 if and only if a = 0 (moάK).

If α = 0 (mod K) then each component is congruent to zero and thesum of the components—that is, σr(nK + L)—is congruent to zero. Con-versely, if σr(nK + L) = a (mod K) as well as σr(nK + L)=α(modIΓ),then, τ(n) standing for the number of divisors of n, we have

[τ(nK + L)l2]a = a (mod K)

since there are τ(nK + L)/2 components each congruent to a (mod K).By Dirichlet's theorem, w and ^ in the proof of Theorem 1 may be

928 V. C. HARRIS AND M. V. SUBBA RAO

taken as primes p and px. Then for nK + L = pply τ(nK + L) = 4. Wemust have 2α = α or α = 0 (mod K).

In the particular case a = 0, conditions (2), (3) and (4) reduce toconditions which Gupta [1] and Ramanathan [2] found to be necessary andsufficient in order that σr(nK + L) = 0 (mod K) for r, n, K and L asabove. Thus we have the remarkable result:

THEOREM 4. Let r, K and L be positive integers with (K, L) = 1and K ^ 3. TΛew σr(nK + L) = 0 (mod iΓ) /or αZZ w ^ 0 if and onlyif σr(nK + L) = 0 (mod ίΓ) /or αM n ^ 0.

REFERENCES

1. H. Gupta, Congruence properties of σ(ri), Math. Student, XIII, 1 (1945), 25-29.2. K. G. Ramanathan, Congruence properties of σr(n), Math. Student, XIII, 1 (1945), 30.3. M. V. Subba Rao, Congruence properties of ΰ(ri), Math. Student (1950), 17-18.

SAN DIEGO STATE COLLEGE

SRI VENKATESWARA UNIVERSITY AND UNIVERSITY OF MISSOURI

FOURIER SERIES WITH LINEARLY DEPENDENT

COEFFICIENTS

HARRY HOCHSTADT

I. Introduction, The following problem is posed and solved in thisarticle. A function H{θ) is defined over the interval (0, π), but is asyet unknown over the interval (—π, 0). Furthermore it is supposedthat the function can be expressed as a Fourier series, with certainconstraints on the coefficients. In particular

H(θ) = - ^ + Σ(αn cos nθ + bn sin nθ)2 n=l

where

aan + βbn = cn , n = 0,1, 2, .

a and β are prescribed constants and the cn a prescribed sequence.The question which can now be raised is whether these constraintsautomatically continue the function into the interval (—TΓ, 0). It willbe shown that under certain conditions the continuation of H(θ) is uniquealmost everywhere.

There are two trivial special case namely if either a or β areallowed to become infinite. In these cases the proper continuation isas an odd or even function respectively.

A different, but equivalent, formulation is the following. Does thedefinition of H(θ) and the constraints on the Fourier coefficients an andbn allow one to evaluate these coefficients? In order to be able to usethe standard integral formulas for the coefficients H(θ) would have tobe defined over an interval of length 2π. Over the interval (0, π) thetrigonometric functions are not orthogonal so that such integral formulasdo not exist. One can show then that an equivalent statement is thatthe nonorthogonal set of functions {sin (nx — tan"1**//?)} is complete inL2(0, π), for I a \ Φ \ β |. The case | a \ = \ β \ requires some additionalstipulations.

One can also formulate a similar problem involving a functiondefined over the interval (0, co), and constraints on the Fourier cosineand sine transforms.

In both of these case one can show that a unique continuationexists in the space of square-integrable functions for \a\Φ\β\. Inthe case of the problem of the infinite interval one can explicitly demon-strate nonunique continuations in the space of nonintegrable functions.

Received October 5, 1961.

929

930 HARRY HOCHSTADT

The proof in both cases is accomplished by reducing the problem tothe solution of a singular Fredholm integral equation of the secondkind. An analysis of the spectrum of the resulting linear operatorshows that the lowest eigenvalue is outside the region of interest.

IL Statement of the theorems.

THEOREM A. Suppose the periodic function H{θ) possesses theFourier series

H{θ) = - ^ + X(an cos nθ + bn sin nθ)

where the Fourier coefficients are linearly dependent. They satisfythe relationship

aan + βbn = cn , n ^ 0.

where a and β are prescribed real constants and the sequence {cn} issquare-summable. If H(θ) is defined as a square-integrable functionover the interval (0, π), there exists a unique (a.e.) square-integrablecontinuation of H(θ) into the interval { — π, 0), provided \a\ φ \β\.

When a = β, one also requires that the function

K{θ) = H{θ) - cr\cJ2 + Σcn cos nθ]1

be such that

oo oo

ΣK < oo, and ^Σt\kn\lnn< oo0 1

where

kn = ('(cot ΘI2)1I2K(Θ) cos nθ dθ .Jo

When a = —β the cot θ\2 is to be replaced by tan θ\2 in the aboveintegral.

Theorem B is a companion theorem to A.

THEOREM B. Suppose the function H(θ) can be represented by theFourier Integral

H(θ) = I (a(ω) cos ωθ + b(ω) sin ωθ)dωJo

where the Fourier cosine and sine transforms are linearly dependent.They satisfy the relationship

FOURIER SERIES WITH LINEARLY DEPENDENT COEFFICIENTS 931

aa(ω) + βb{ώ) — c(ω) , ω ^ 0

where a and β are prescribed real constants and the function c(ω) issquare integrable. If H(θ) is defined as a square integrable functionover the interval (0, oo) there exists a unique (a.e.) square integrablecontinuation of H{θ) into the interval (— oo, 0), provided \a\ Φ \ β \.

When a — β one also requires that the function

K(θ) = H(θ) - — [°°c(ω) cos ωθ dθa Jo

be such that

\ k\ω)dω < oo, and \ \ k(ω) | In ω dω < oo .Jo Jo

where

k(ω) = \~θ~ll2K{θ) cos ωθ dθ .

When a = — β, θ~112 is to be replaced by θ1'2 in the above integral.Equivalent formulations of these theorems are the following.

THEOREM A'. A function H(θ) in L2(0, π) can be represented inthe form

H{θ) = ΣK sin (nθ + φ)

where φ is a fixed phase angle. For φ = ±τr/4 one must impose ad-ditional restrictions on H{θ) as in Theorem A.

THEOREM B'. A function H(θ) in L2(0, oo) can be represented inthe form

H(θ) = [°k(ω) sin (ωθ + φ)dωJo

where φ is a fixed phase angle. For φ = ±ττ/4 one must impose ad-ditional restrictions on H(θ) as in Theorem B.

However the former formulation is preferable because that is thedirect form in which the theorems are proved.

ΠL Reduction of the proofs to the analysis of integral equations*One can in the ensuing analysis replace the cn by zero without loss ofgenerality since the general expansion can be rewritten in the followingform after an is eliminated.

932 HARRY HOCHSTADT

H(θ) - 1/αΓA. + £C ncosnθ\L 2 i J

- sin %0

Let h(—θ) denote the continuation of H{θ) in the interval (—π, 0),and αΛ, &„ denote the Fourier coefficients of the resultant function. Then

and let d% and en be defined by

ί>« {£}•"»-«{£}•Thus one can solve for the corresponding integrals for h(θ) and

!:»<•» U S } - * - = * - " ; } •From these two equations the unknown coefficients an and bw can beeliminated by use of the relationship

aan + βbn = 0 .

It follows that

(1) I h(θ)(a cos nθ — β sin nθ)dx — π(—adn — βen), n — 0,1,Jo

One can now multiply the above equation first by a cos ny and thenby β sin ny and take the difference of the resultant equations, to obtain

a* + / g 2 Vh{θ) cos n(θ — φ)dθ + a2 ~ ^ Γ&(0) cos n(θ + φ)dθ

— α/51 h(θ) sin ^(^ + φ)dθ = (ττ(—αdπ — βen)(a cos nφ — β sin ^φ) .

JoOne can now apply the summation formulas

s i n ( i V - — λx1 N V 2— + Σ cos nx =

Σsin^=|-cot^--

2

cos (JV— —)xV 2/

to the above equation and then pass to the limit as N tends to infinity.One then obtains the integral equation

(2) h(φ) - — [πh(θ) cot θ + φ dθ = f(φ)

where

. _ 2α/9

a2 + β2

f(Φ) = —i — ] ~a ° + Σ (—adn — β θ ( α cos nφ — β sin nφ) \ .

To convert the Fourier integral case to an integral equation onedefines d(ω) and e(ω) by

and proceeds in a similar fashion as in the previous case. There is analternative procedure. The period is changed from π to T by a formalchange of variable and by a passage to the limit as T tends to infinityone obtains

(4)7Γ Jo ΰ -\- φ

where

λ

α2 + β2

___±—__ \ (—ad(ω) — βe(ω))(a cos ωφ — β sin ωφ)dω .

IV* Analysis of the integral equations* The integral equationscorresponding to both problems are singular integral equation of theFredholm type of the second kind. It will be shown that both equa-tions have unique solutions in the space of square-integrable functionsprovided that the eigenvalue parameter λ satisfies

| λ | < 1 .

But since

λ _ 2aβα2 + /92

and the latter function is bounded by unity it is evident that the in-

934 HARRY HOCHSTADT

tegral equations alway have unique solutions in the space of square-integral functions. The case | λ | = 1 will be treated separately.

Equation (4) is discussed in detail in [3], and the same method canbe adopted for equation (2).

We now consider equation (2) and expand the kernel in terms ofan orthonormal system of functions over the interval (0, π). We findthat with the kernel we can associate the quadratic form

Σ antkxnxkn,k=l

where the antk are given by

ank = —[*[* cot θ + φ sin nθ sin kφ dθ dφπ JoJo 2

n + k evenOM / \n+kl

= j^ι± v—i—j_ =

n + k4

n + kn + k odd ,

if the selected orthonormal system is {(2/ττ)1/2 sin nθ}.We now consider the analytic function

F(z) = Σ a?^""1

1

and suppose {xn} to be a square-summable sequence. A direct calculationshows that

Γ zF\z)dz = i Σ a«,krn+kxnxk, 0 ^ r < 1 .

J-r 2 n.k=l

One can also show that the quadratic form is bounded over thespace of square summable sequences.

I Γ zF\z)dz - I Ϋr*e**F\reiφ)dφ

^ A*\ F(reiφ) \2dφ = ^[{Σx^-1 cos (n - l)φ)2

Jo Jo

+ (Σxnrn^ sin (n - l)φ)2]dφ = πΣx2

nr2n

^ πΣxl .

By letting r tend to unity one finds

Σ

In order for equation (2) to have a unique solution in the space of

square-integrable functions it is necessary and sufficient that the quad-ratic form

Q(x) = f>l - - £ - Σ αi Z7Γ n,k=i

be positive definite. We see that this form can be written as

Q(x) = lim Γ— (V1 F(retφ) |2 dφ - —Γ zF\z)dz\ .r- l L7Γ JO π J-r J

This expression must be real, and writing

F(z) = R(r, φ)emr'^

we obtain

Q(x) = lim Γ—(V2i22(r, φ)dφ - — (V.K2(r, cp) sin {2θ(r, φ) + 2φ]dφ\ .r-i Lπ Jo 7Γ Jo J

Evidently this is positive definite if | λ | < 1.The preceding type of argument was first used by Fejer & F. Riesz

[1], to discuss the bounds of such operators. But one can show stillmore, namely that the bound of the operator is not attained for anyvector x. If it were Q(x) would vanish, in which case

sin {20(1, ψ) + 2φ} = 1 , a.e.

In this case the real part of the function z2F\z), is a harmonic function,which vanishes a.e. on | z \ — 1. Such a harmonic function can be re-presented by a Poisson Integral [2]. In follows therefore that since itvanishes a.e. on | z \ — 1 it must vanish identically and it follows thatthe function zF(z) must also vanish identically. Therefore Q(x) doesnot vanish for any x. One can infer from this that the homogeneousintegral equation has only the trivial solution, so that the inhomogemeousequation will have a unique solution provided a solution exists even inthe case | λ | = 1. But the existence of a solution depends on thenature of the inhomogeneous term. This case will be discussed in thenext section.

It follows that for | λ | < 1 the integral operator is a contractionoperator so that the solution can be obtained by successive iterations ofthe operator.

A similar analysis can be carried out for equation (4) using as anorthonormal set over (0, oo) Laguerre polynomials. The rest of theanalysis is similar and details may be found in [3]. However one canapproach this problem also by the use of Fourier integrals. This ana-lysis can be found in Titchmarsh [4]. The substitutions

936 HARRY HOCHSTADT

φ = e\ θ = β*, e^h^) = Φ(rj)

e() = Ψ{ξ)

reduces equation (4) to the form

cosh — (v — ξ)2

Let

τi

and it is known that

2

One finds immediately that

coshi coshπω '

F(ω) =λ

cosh πω

so that

(2ττ)1 / 2 -l-o1 -

COSh 7ΓO)

From the expression it is evident that the integral equation need nothave unique solutions. The solutions of the homogeneous equation mustbe of the form evv, where v is a zero of cos (πv) — λ. Thus equation(4) has unique solutions in the space of square-integrable functions, butis only determined to within a nonintegrable term of the form cyv~112,c being arbitrary.

V The case | λ | = 1. When | λ | = 1 we have either a = β ora = — β. We will consider the case a = β in detail and the other casecan be reduced to this one by replacing θ by π — θ. The given functionH{θ) is to be represented in the form

H(θ) = i χ (cos nθ - sin nθ)1

and we introduce the function

f(z) = Σbnzn = u{r, θ) + iv(r, θ) .

1

Evidently

H{θ) = w(l, 0) - v(l, θ) 0 < 0 < π .

Let Z7(r, 0) be a harmonic function defined by

U(r, θ) = w(r, 0) - v(r, 0) ,

whose conjugate harmonic function is given by

V(r, θ) = v(r, θ) + u{r, θ) .

Since the bn are taken to be real u will be even and v will be odd inS. Then

l, θ) = H{θ) 0 < θ < π

V{1, θ) = H{-θ) -π < θ < 0 .

In order to determine the continuation of H(θ) into the interval (—π, 0)it is necessary to determine U(r, θ) for all θ. We now define thefunction

F(z)= U+ίV

and introduce the function

\l/2

— z

with the boundary values

G(eiΘ) = (cot ΘI2)112 , 0 < θ < 7Γ

= -i(cot -0/2)1/2 , -π < 0 < 0 .

The function G(z)F(z) ~ T(z) is an analytic function whose real part isdefined for the whole boundary.

ReίG{z)F{z) = (cot θβY'H{θ) , 0 < 0 < π

= (cot -θfeY'Ήi-θ) , -7Γ < θ < 0 .

Thus T{z) is explicitly given by

T(z) = %e-\

= A Γ(cot ΘI2)1I2H(Θ) cos 0π Jo

938 HARRY HOCHSTADT

and c is a real, but otherwise arbitrary constant of integration. F(z)is now fully determined and it follows that

U(l, θ) = Re^ζL = H(θ) , 0 < θ < π

= -(tan -0/2)1'jc + | χ sin nθ\, -π < θ < 0 .

Here J7(l, θ) is not uniquely specified, but if one requires that 17(1,0)be square integrable the constant c must be set equal to zero. Fur-thermore it is not enough to require

Σkl < oo ,

but one also needs

Σ\ kn I In n < oo

in order for

Γ tan θ/2[Σkn sin nθYdθ < oo .Jo

The Fourier integral case be treated in an analogous fashion or byformal limiting processes.

VI Proof of Theorems A and B To prove Theorem A it is stillnecessary to show that the periodic function, which is given by h(—φ)for — π < φ < 0 and H(φ) for 0 < φ < π has the required properties*From the definitions of the coefficients dn and en it follows that

1 °°

—a?d0 + Σ0*2^n cos nφ — β2en sin nφ2 i

+ aβ en cos nφ — aβ dn sin nφ)

S *denotes the principal value of the integral. One can by the use of

this summation formula now rewrite equation (2) to read

.H(φ) + -V h(-θ)cot°^dθ

dθ = 0 .( H ( g ) c o t2π Jo 2

To complete the proof one merely observes that

H(Φ) = -7r + Σ(α* cos nφ + bn sin nφ)2 i

KΦY= -y- Σ(« cos ίiφ - &re sin

Then the previous equation reduces to

α2-^- + α 2 | χ cos nφ - /32i>% sin2

Σfn cos w0 —1

which evidently shows that

aan + βbn = 0 , w ^ 0 ,

and thus completes the proof of Theorem A.The proof of Theorem B is completely analogous and will therefore

be omitted.The statements of the theorems can be considerably strengthened

if one assumes that the original function H(θ) defined for 0 < θ < π iscontinuous and bounded and the {cn} are such that the inhomogeneousterms in (2) and (4) are also continuous and bounded. In this case itfollows from the existence of the Neumann series that the functionh{θ) is also continuous and bounded for 0 < θ < π. Then the resultantperiodic function is continuous and bounded at all points with the ex-ception of points of the form nπ.

REFERENCES

1. L. Fejer, and F. Riesz, ϋber eίnίge funktionentheoretische Ungleichungen, Math. Zeit,1 1 (1921).2. G. M. Golusin, Geometrische Funktίonentheorίe, Deutscher Verlag der Wissenschaften,1957. Chapter IX.3. W. Schmeidler, Integralgleichungen mit Anwendungen in Physik und Technik, Aka-demische Verlagsgesellschaft, (1955), 485.4. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd Ed; Oxford(1948), 310.

POLYTECHNIC INSTITUTE OF BROOKLYN

A CHARACTERIZATION OF C{X)

KENNETH HOFFMAN AND JOHN WERMER

It is a classical fact that there exist harmonic functions u in theunit disk with conjugate harmonic function v such that u has continu-ous boundary values on the unit circumference, while v does not. Letus restate this fact as follows:

Denote by Ao the algebra of functions analytic in | z | < 1 withcontinuous boundary values on | z | = 1 and write Re Ao for the space ofall real parts of functions in Ao. Then we may say: there exists aharmonic function u in | z | < 1 with continuous boundary values suchthat u does not lie in ReA0. On the other hand, u is certainly a uni-form limit of functions in ReA0 on \z\ •= 1, for all finite real trigono-metric polynomials on \z\ — 1 are in ReAQ. Thus we see: ReA0 is notclosed under uniform convergence on | z \ = 1. In this paper, we shallshow that this phenomenon is a special case of a very general propertyof algebras of functions.

Let X be a compact Hausdorff space and C(X) the algebra of allcontinuous complex-valued functions on X. Let A be a complex linearsubalgebra of C(X) such that

(1) A is closed under uniform convergence;(2) A contains the constant functions;(3) A separates the points of X.

We write Re A for the set of functions Ref with / in A, that is, forthe set of real parts of the functions in A. Clearly Re A is a (real)vector space of real-valued continuous functions on X. The purpose ofthis paper is to prove the following.

THEOREM. / / Re A is closed under uniform convergence, thenA = C(X).

COROLLARY 1. If Re A contains every real-valued continuous func-tion on X, then A = C(X).

COROLLARY 2. (Stone- Weierstrass) If A is closed under complexconjugation, then A = C{X).

Corollary 1 is an evident consequence of the theorem, and Corollary2 follows upon observing that, if A is closed under complex conjuga-

Received October 25, 1961. The research of the first author was supported in part bythe Office of Scientific Research, Air Research and Development Command, under ContractNo. AF 49(638)-1036. The second author is a Fellow of the Alfred P. Sloan Foundation.

941

942 K. HOFFMAN AND J. WERMER

tion, Re A is simply the collection of real-valued functions which arecontained in A. The proof of the theorem proceeds by reducing it tothe case when A is anti-symmetric, i.e., every real-valued function inA is constant. Let us first settle this case.

LEMMA. If Re A is closed and A is anti-symmetric, then the spaceX contains not more than one point.

Proof. Suppose that X contains at least two points. Fix a pointx0 in X, and let (ReA)0 be the class of all u in Re A with u(x0) — 0.

Suppose u is in (Re A)o. Let / be a function in A such that u =Ref. Since the constants are in A, we may assume that v = Imfvanishes at xQ. Since v = Re{—if), we then have ve(ReA)0. Nowgiven u, the function v in (Re A)o such that (u + iv) is in A is uniquelydetermined. For, if vr is another such function, (v — vf) is a real-valuedfunction in A. Since A is anti-symmetric v — vr is constant, and thecondition v(x0) = v'(x0) = 0 tells us that v = v\ Put v = Γw.

Then Γ is a linear transformation of (/2eA)0 into itself. Since weare assuming that Re A is closed under uniform convergence, (ReA)0 isa Banach space with the norm

11 u 11 = sup I u I .

We claim that T is a bounded operator on this Banach space. To provethis, it will suffice to show that the graph of T is closed. Suppose wehave a sequence of elements un in (Re A)o such that un —• u and Tun —»vuniformly. Then the functions (un + iΓ%n) lie in A and converge uni-formly to (u + iv). Thus (u + iv) is in A, and since it is apparentthat v(xQ) = 0, we have v = Tu. We conclude that Γ is bounded.

Since X contains at least two points, we may choose a nonconstantfunction g = s + it in A such that #(#0) — 0. Let R denote the rectan-gle in the complex plane defined by

Then g(X) is a compact subset of R. Since g is nonconstant, we cannothave β = 0 or t — 0. In particular, there is a point xx Φ x0 in X suchthat I ίfo) I = || ί ||. Let z0 = ^(^), so that ^0 is a boundary point of R.

Fix any integer N> 0. There exists a conformal map φ of theinterior of R onto the interior of the rectangle RN\

such that ^(0) = 0 and θ(z0) = iN. Since R and RN are rectangles, theconformal map φ extends to a homeomorphism of the boundaries of Rand RN. In particular, φ is a uniform limit of polynomials on R. There-

A CHARACTERIZATION OF C(X) 943

fore, the function h = φ(g) is in the algebra A, and h(x0) = 0(0) = 0.If h = u + iv we have

IMI = N.

Since JV was arbitrary and v = Tw, we have contradicted the fact thatT is bounded. Thus X cannot contain more than one point.

Proof of theorem. A theorem of Bishop [1] states the following.If A is a subalgebra of C(X) satisfying (1), (2), (3), there exists a parti-tion P of the space X into nonempty disjoint closed sets, such that

( i) for each S in P the algebra As, obtained by restricting A toS, is anti-symmetric;

(ii) As is a uniformly closed subalgebra of C(S);(iii) the algebra A consists of all continuous functions / on the

space X such that the restriction of / to S is in As for each S in thepartition P.

Glicksberg [2] proved that we may also arrange that(iv) if S is a fixed element of P and T is a closed subset of X

disjoint from S, there exists a function g in A such that

Hffll ^ 1 , g = 1 on S, \g\< 1 on T.

Actually, (ii) is a consequence of (iv). What we shall show now isthat (iv), together with the assumption that Re A is closed, implies thatReAs is uniformly closed for each set S in the partition P. This willprove the theorem. For As is an anti-symmetric closed algebra on thespace S, and the above lemma shows that S consists of one point. By{iii) we then have A — C(X).

Fix S in P. We show that ReAs is closed. We first assert thefollowing. If / 6 A and ε > 0, we can find F e A such that

(4) suv\ReF\ ^ sup|Λβ/| + 2ε , and ReF= Ref on S .X S

Let Ω be the region in the w-plane (w — u + iv) defined by

w I < 1 , — ε < ^ < ε .

Let τ be a conformal map of \z\ < 1 on Ω with τ(0) = 0 and r(l) = 1.Choose δ > 0 such that τ maps | z \ < δ into | w \ < ε. Choose a neigh-borhood U of S in X with

I Λ β / | ^ sup I jβe/ | + ε , on U.s

By (iv) above there is a g e A such that || g \\ ^ 1, g = 1 on S, | # | < 1on X — ί7. Choose a positive integer n large enough that | gn | < δ on

944 K. HOFFMAN AND J. WERMER

X- U. Put h = τ{gn). Then h e A, h = 1 on S, and \Imh\^ε on allof X. Also I Reh| < ε on X — U and \Reh\<Ll on all of X. Nowdefine F = /A. Then F e A and

ReF = RefReh- Imflmh .

Therefore

( 5 ) ReF= Ref on S

(6) iΛeJFΊ ^ (sup|Λe/| + ε) + ε, on Us

(7) | / ί e F | ^ ε + ε, on X- U.

In particular, F satisfies (4). (For (6) and (7) we have used | | / | | ^ 1.)We finish the proof with a standard closure argument. Let R&

denote the subspace of Re A consisting of all functions in Re A whichvanish on S. With norm given by maximum modulus over X, Re A i&a Banach space, and Rs is a closed subspace. The quotient space Q —ReA/Rs is therefore complete in the norm

\\Ref+ Rs\\ = mΐ\\ReF\\ , ReF=Ref on S.F

But by (4)

sup]iίe/ | = inf \\ReF\\ , ReF= Ref on S .

We conclude that ReAs, which is clearly isomorphic to Q, is completein the maximum norm on S. We are done.

The theorem of this paper was proved independently by H. Rossiand H. Bear.

BIBLIOGRAPHY

1. Errett Bishop, A generalization of the Stone-Weierstrass theorem, to appear.2. Irving Glicksberg, Measures orthogonal to algebras and sets of anti-symmetry, to appear.

THE BEHAVIOR OF SOLUTIONS OF ORDINARY,SELF-ADJOINT DIFFERENTIAL EQUATIONS

OF ARBITRARY EVEN ORDER

ROBERT W. HUNT

Introduction* It is the purpose of this paper to establish someproperties of the zeros of solutions of ordinary, self-adjoint differentialequations of arbitrary even order of the form

(1) [r(x)yM]M + (-iy+1p(x)y = 0

where r(x) > 0, p(x) > 0, and both coefficients are continuous on [α, oo).Of particular concern is the existence of a nontrivial solution of (1)which satisfies one of the following sets of two-point boundary con-ditions

(2) y(a) = y'(a) = . . . = y^\a) = 0 = y(b) = y\b) = . . . = y^\b)

(3) y(a) = y\a) = ... = y^\a) = 0 = y1φ) = y[(b) = . . . = y[^\b)

where yx{x) = τ(x)y{n)(x), a notation which will be continued throughoutthe discussion, and b > a.

Recently the special fourth-order case (n — 2) has been investigated ex-tensively by W. Leighton and Z. Nehari [10], by H. M. and R. L. Sternberg[13], by H. C. Howard [8], and by J. H. Barrett [2, 3, 4]. In the presentpaper some of the methods of Barrett [2,4] are extended to the generalcase; and, in so doing, some of the arguments used for n = 2 are simplified.

W. T. Reid has recently announced [12] a general discussion includ-ing the above types of zeros of solutions of quasi-differential equations ofeven order of which (1) is a special case. Reid discusses related eigenvalueinequalities and his methods are variational in nature and assume somebasic results of the spectral theory for boundary problems that havebeen established earlier in the study of the calculus of variations.

This discussion, which generalizes Barrett's methods, has the ad-vantage that only fairly well-known properties of matrices and differ-ential equations are used. Furthermore, and most important, a con-siderably stronger criterion for the existence of a non-trivial solutionsatisfying (2) (see Theorem 4.3) and of one satisfying (3) (see Corollary5.1) is established by utilizing the simple form of (1). Then two com-parison theorems, established by an application of Reid's variationalresults [12], extend these stronger results to the general self-ad joint

Received May 29, 1961. This paper is part of a thesis submitted to the faculty of theUniversity of Utah in partial fulfillment of the requirements for the degree of Doctor ofPhilosophy. The author expresses thanks to the thesis director, Professor John H. Barrett.Presented to the American Mathematical Society, April 22, 1961.

945

946 ROBERT W. HUNT

case, i.e., the differential equation of the form

(4) [rn(x)V(nΎn) + Σ P o K - l ^ + V ^ ) ^ ^ = 0 ,

where r^x) > 0 for i = 0,1, 2, , n, and all of the coefficients arecontinuous on [a, co). This extension is discussed in §6.

For the sake of completeness, there is developed in the first sectiona canonical representation of (1) as a system of two first-order matrixequations as given by H. Kaufman and R. L. Sternberg [9], as modifiedby Barrett [4], and as modified here by a method suggested by aproblem in [6, problem 19, p. 206]. This system is of the form

Y' = E(x)Z , Zr = -F(x)Y,

and is so designed that a singularity of the ^th-order matrix Y(x) atx = b gives the existence of a nontrivial solution of (1) satisfying theconditions (2); and a singularity of Z(x) at x = h gives the existence ofa nontrivial solution of (1) satisfying the conditions (3). These propertiesare discussed in § 2. In § 3, the determinants of Y(x) and Z{x) areshown to satisfy certain second-order, self-adjoint differential equationswhich generalizes a result for the case n = 2 due to Leighton andNehari [10] and Barrett [2], In §§4 and 5, conditions for the existenceof nontrivial solutions satisfying the conditions (2) and (3) are discussed.

l A matrix differential system related to (1)» This discussionparallels [4] with a slightly different modification suggested by a prob-lem in [6, page 206]. Let y(x) be any solution of (1) and let

y1(x) = r(x)y{n)(x) ,

a notation which will be used throughout the discussion. Define thewth-order column vectors a(x) and a(x) by

a(x) = (α.) = (y«-»), a(x) = {&,) - ( ( - 1 ) % ^ ) >

where i — 1, 2, , n. Then a(x) and ά(x) satisfy the system

a! = Ba + C(x)ά

a' = A{x)a - B*ά ,

where A(x), B, and C(x) are n x n matrices defined as follows, (i denotesthe row index and j the column index).

A(x) = (ai3{x)) ,

where

Ma) = (~l)n+1P(%), a,ij(x) = 0 for i Φ 1 or j Φ 1 .

B = ψij), where bid = 1 for j = i + 1, bi5 = 0 for j Φ i + 1 .

C(x) - (ci3(x)) ,where

THE BEHAVIOR OF SOLUTIONS OF ORDINARY, SELF-ADJOINT 947

e»nO*O — (—ϊ)nlr(x), ciά(x) = 0 for i φ n or j Φ n .

Next, consider the nxn matrices D(x) and D(x) which satisfy

D' = BD, D(a) = In and D' = - B * A J5(α) - J. ,

respectively, where In is the w x n identity matrix. For simplicity inthe remainder of the paper, and with no loss in generality, let a = 0.Now, the equations for D(x) and D(x) can be solved to give

D(x) = (d^x)), where diS(x) = 0 for i > j and

dij(x) = xύ~ιlU - ϊ)\ for i ^ j .

D(x) = (d^ (^)), where d^-fa) Ξ 0 f or i < i and

diό{x) - (-ly-V-^/ίi - i ) ! for j £ i .

Now, let /δ(a?) and β(x) be ^th-order vectors defied by the relations

a(x) = D{x)β{x), ά(x) = ( - l ) 5(a?),β(a?) .

Substitution into (5) then yields

β> = £7(0;)^( 6 ) /3' = - F ( x ) ^ ,

where E(x) and F(ίc) are w x n matrices defined as follows.

E(x) = (βiiίa?)), where

eiά(χ) = (—ly+V*-*^/^ - i)!(w - i ) ! φ θ ,

^(») = (/«(*)), where /4J(a?) = x^v&W - l ) ! ( i - 1)!for i ^ i and /<,.(») = /<,.(») .

Thus, ^(a;) and F(x) are symmetric, positive semi-definite matrices.D{x), E(x), and F(x)t are generalizations of matrices used by Barrett[4] for n = 2.

Let {Ui(x)}f i = 1, 2, 3, •• , n be a set of solutions of (1) whichsatisfy the boundary conditions

[UiixW'-v = 0 for x = 0 and ί, i = 1, 2, , n .

N.iίa?)]0'"" = δϋ for x = 0 and i, i = 1, 2, , w

and M<i:ι(ίB) = r(a;)^Λ)(ίc) .

Now, denote the ^th order vector with components uij~1](x), j = 1,2, , n, by aUt(x) and the ?ιth order vector with components(-l)%ί?f i }(a0, ί = 1, 2, , n, by αW£(x). Then define

(8) i8tt4(αj) = D - ^ K / x ) , ^(a?) = ( - l ) ^ - 1 ^ ) ^ ^ ) , i = 1, 2, . . , n .

THEOREM 1.1. Lβί F(α?) be an nxn matrix whose columns (in the

948 ROBERT W. HUNT

order i = 1, 2, , n) are the βUi(x) and let Z(x) be an nxn matrixwhose columns are (in the order i = 1, 2, , n) the βH(x). Then Y(x)and Z(x) satisfy

(9) Y* = EZ, Z' = -FY.

Proof. This theorem can be verified by direct substitution.

2. Relations between the system (9) and equation (1) First, twotypes of zeros of solutions of (1) are defined as follows.

DEFINITION 2.1. The number ^(0) is the smallest number b on(0, oo) such that the boundary conditions

(10) y(0) = τ/'(0) = ... = j/<-»(0) - 0 - y(b) = y\b) = ... = y«-*Q>)

are satisfied nontrivially by a solution y(x) of (1). This is the type ofboundary problem considered in [2,4, 12, 13].

DEFINITION 2.2. The number μ^O) is the smallest number b on(0, oo) such that the boundary conditions

(11) 7/(0) - y'(0) = ... = ί,<-»(0) = 0 = yλ{b) = y[φ) = ... = y[^\b)

are satisfied nontrivially by a solution y{x) of (1). This is a generaliza-tion of the type of condition first used by Barrett [2,4] and Howard[8] as an intermediate condition to (10).

THEOREM 2.1. A number b ε (0, oo) is the smallest number on thatinterval for which det Y(b) — 0 if and only if b — 3 (0) for (1). Anumber b ε (0, oo) is the smallest number on that interval for whichdet Z(b) = 0 if and only if b = μλ(0) for (1).

Proof. Note first of all that det Y= W[ulf ---,un] and det Z(x)— W[u1Λfu2ιl, * ,un>1], where W turns out to be a Wronskian in eachcase. The former is true since

Y(x) = D-\x)Mx) ,

j*(x) being the matrix of W[ulf u2, , un], and det D(x) = 1. To verifyt h a t det Z(x) = W[u1Λ,u2tl, •• , / M W I 1 ] , observe that

jt(x) being the matrix whose columns are the ccUi(x) in the order i = 1,2, ,w. But det (-l)nj?{x) = W[u1Λ, u2ιl, . , un,ί] and det D(x) = l.Using the definition of the functions u{(x), the result now follows.

2* Second-order, self-adjoint, linear differential equations whichliave det Y(x) and det Z(x) as solutions* To simplify the notation, let

det Y(x) = σ(x) , det Z{x) = p(x)

in the remainder of the discussion. Note that σ(0) = 0 and ^(0) = 1.Consider, first, the following lemmas.

LEMMA 3.1. Let σ(x) denote the Wronskian, W[uu u2, , u^](n > 1) where the functions ut(x) are as defined in § 1 and (7). Thend{x) Φ0 on (0, oo).

Proof. Assume that σ(x) has a zero on (0, oo). Then there is asolution y(x) of (1) which is a linear combination of u^x), u2(x), •••,un-x{x) and a minimum value x = b on (0, oo) such that y(0) = y'(0) =. . . = y<»-»(0) - ^- 1 } (0) - 0 and y(b) = y\b) - = y (-«(δ) - 0.

Let ?/(#) have m distinct zeros on (0, b) (m ^ 0), the first such zeroΐ>eing x = c on (0,6]; and suppose, without loss of generality, that#(#) > 0 on (0, c). Then, because of the conditions on y(x) at x — 0,^(0) ^ 0. Now, noting the condition on y(x) at α? = 0 and x = b,Kolle's theorem can be applied to y{x), y'(x), •••, yίn~2)(x) in turn to givethe following information.

y\x) has at least m + 1 distinct zeros on (0, 6).y"(x) has at least m + 2 distinct zeros on (0, 6).

(»-!)(/») has at least m + n — 1 distinct zeros on (0, 6).^(cc) has at least m + n — 1 distinct zeros on (0, b).y[{x) has at least m + n — 2 distinct zeros on (0, 6), the first one

of which can be chosen to the right of the first of the abovem + n — 1 distinct zeros of yλ(x).

Ui(x) has at least m + n — 3 distinct zeros on (0, 6), the first oneof which can be chosen to the right of the first of the abovem + n — 2 distinct zero of y[(x).

yίn~1](x) has at least m distinct zeros on (0, b) if m Φ 0, the firstone of which can be chosen to the right of the first of theabove m + 1 distinct zeros of y[n~2){x).

(If m = 0, the last statement is replaced by the statement thatηjίn-\x) Φ 0 on (0, 6).)

Then, from the last statement for m Φ 0, y[n){x), and hence y(x),has at least m distinct zeros on (0, b) since ^"""(O) = 0. If m = 0,y(x) has no zeros on (0, b). But y(x) has exactly m distinct zeros on<0, b). Thus, each of the functions yίn~1](x), •• fyϊ(x),vΊ(x) must haveexactly the number of distinct zeros on (0, b) as given in the precedingstatements since each has at least that number of zeros and if any

950 ROBERT W. HUNT

one had more distinct zeros, it would follow that y(x) had more thanm distinct zeros. Also, all of the distinct zeros of y(x) are simple.

Next, y^'îx) — (—l)nl p(t)y(t)dt. Consider the case where n is even.

Then ylîx) > 0 for x on °(0, c). Thus, yίn~2)(x) must begin at x = 0with zero slope, have positive slope on (0, c), and have a zero beforeits slope has a zero (since for m Φ 0 the first zero of y^'îx) is to theright of the first zero of y[n~2)(x) on (0, oo) and for m = 0, y^'îx) Φ 0on (0, b) by the above considerations). This is possible only if j/ίn"2)(0)< 0. Then y[n~*\x) must begin at x = 0 with negative slope and havea zero before its slope has a zero which implies that yίn~d)(0) > 0.Iteration of this argument gives ^(0) < 0, a contradiction. For thecase in which n is odd, this contradiction is obtained similarly. Thus,σ(x) ^ 0 o n (0, oo).

LEMMA 3.2. Let p(x) denote the Wronskian, W[ulιlfu2ιl, " ,un-1Λ](n > 1) where uJtl(x) = r(x)Ujn)(x) and the functions u{{x) are as inLemma 3.1. Then p{x) > 0 on [0, oo).

Proof. This result is proved in the same way as the precedinglemma. p{x) > 0 on [0, oo) since /?(0) = 1.

LEMMA 3.3. The solutions u^x), as previously defined, satisfy thefollowing set of n(n — l)/2 identities (n > 1):

(12) m-m^xWZTΓHx) - u%){x)utϊ™-v{x))\ = 0 ,ra=0

i = 1, 2, , n — 1 and j — 1, 2, , n — i »

Proof. The lemma is established by an induction on the followinggeneral observations:

(ruln)){n) + (-ly+'pUi = 0 , (ru'^j)™ + (-l)n+1puί+j = 0

and hence uJtruft,)™ - ui+j(ruln)){n) = 0; that is, uiu(^jΛ - uί+ju[ni = 0.

Then n integrations by parts from 0 to x give

+ Σ (-l)"+HtίϊKr"-u + (-l)n+1[uil)uiΛ = 0 ,m=0 JO

from which the lemma follows.Next, note that the function {r(x)σ'{x))f is as follows:

(13) (r(x)σ'(x))> - Dx{x) + D2(x) ,

where Dλ(x) is an nth order determinant whose first n — 1 rows are

(u[j)u{

2

j) up), j = 0, , n — 3, n — 1, and whose last row is (u1Λ, u2tl

• w».i); and D2(x) is an wth order determinant whose first n — 1 rowsare the same as those of Dτ{x) except that j = 0, , w — 2, and whoselast row is (u'1Λ, u'2tl u'n>1).

LEMMA 3.4. (r(#)0"'(#))' com 6β written as

(13') (r(α)e7'(aθ)'

/% other words, the two determinants on the right-hand side of (13) areidentically equal.

Proof. Let Witi+j denote the sub-Wronskian of W[ulf u2, , un]which is obtained from the latter by deletion of the ith and (i + i)thcolumns and the last two rows. Then, using the equations (12), consider

m = 0{*>«ft7"-1> - « ίτ}<r"- u ] = o ,

for i = 1, 2, , % — 1 and j = 1, 2, , n — i. Summing these n(n — l)/2identities yields

i=l j=l m=0

Now take m = n — 1. This gives

which is exactly the first determinant on the right-hand side of (13) ifn is odd and the negative of the determinant if n is even. This canbe seen by expanding the determinant by Laplace's development basedupon minors of the last two rows.

Next take m = n — 2. Then, as above, this gives the negative ofthe second determinant on the right side of (13) if n is odd and thedeterminant if n is even.

The result (13') will now follow if it can be shown that

n—1 n—i n—3

In particular, this holds for each m fixed between 0 and n — 3 asfollows. Fix m so that 0 ^ m ^ n — 3. Then consider the identicallyzero determinant (two identical rows)

952

ux

u[

VΛ.in-m-D»1.1

u2

ui

iry^(n-m-l)

ROBERT W. HUNT

••• u.

••• <

• %iB-3 )

• Ίjfy '

= Σ,Σ,[(-irH+2n-1wi,i+3{uruiι-Jτ1) - ttίTXr--1')],

the expansion being by Laplace's development based upon minors of thelast two rows. But this expansion is either equal to the above with afixed m or the negative of the above with a fixed m. Thus, the asser-tion holds; and (13') follows.

THEOREM 3.1. The function σ(x) — det Y(x) is a solution of

(14) (rσ'lσ2)' + 2r(δ/α3)σ - 0 ,

on (0, co), where δ(x) is a determinant of order n — 1 whose rows are(u[j)u{

2

ά) n where j = 0, , n — 3, n (n > 1).

Proof. Using (13'), the differential equation (14) can be writtenin the form (2lσ)(D1σ — rσ'σ' + rSσ) — 0. Now, if it can be establishedthat the left hand side of this equation is identically zero on (0, oo), (14)will be established. To show this, consider the following identicallyzero determinant (can be verified by the use of induction and expansionby minors) and its Laplace development based upon minors of the firstn — 1 columns.

(15)

u[ ui

0

u[

0

ui

< - i

%iri2)

oo

0

0

u2

ui

ui'

Λi(n-l)Wn-l

0

0

K

u.(n-3)

( ) ( ) = 0 .

The right hand side of this equation is exactly the term which was to

be shown identically zero. This completes the proof of the theorem.An analogous result to Theorem 3.1 for ρ{x) = det Z(x) is given as

follows.

THEOREM 3.2. The function ρ(x) = det Z{x) is a solution of

(16) (p'lPP2)' + (2lp)(δjps)p = 0 ,

on (0, oo), where

(17)K-1,1

η,(n)

4. Conditions for the existence of /*i(0)* This first two theoremsof this section generalize results for the case n = 2 by Barrett in [2,4]. These results could, also, be obtained by specialization of Reid'swork in [12]. Theorem 4.3, on the other hand, generalizes Barrett'sLemma 2.6 in [2], gives an improvement over Theorem 4.2, and givesa condition for the existence of μx(0) which improves the specializationobtained from [12].

THEOREM 4.1. // ^(0) exists, then μ^O) exists and 0 <

Proof, det Z(0) = 1 and thus there is a maximum value x = b on(0, oo] such that Z(x) is nonsingular on [0,6). On [0,6), h t K(x) =Y(x)Z~\x) so that K(x) is symmetric and

(18) K\x) = E(x) + K(x)F(x)K(x), K(0) = 0 .

K(x) has been discussed and utilized in [1] and [11], as well as [4]where Barrett has used it for the case n = 2. Now let f = (!<) be anwth order, constant, column vector. Then, using (18),

ξ*E(t)ξdto

{-1)n~lξΛn - 2 ) ! Jd t '

and hence K(x) is positive definite on (0, 6). Thus, det K(x) =det Γ(α?)/det Z(α?) is positive and Y(x) is nonsingular on (0, b). This impliesthat 2 (0) does not exist on (0, b) and the result of the theorem follows.

954 ROBERT W. HUNT

THEOREM 4.2. / / I x2{n~1)p(x)dx = oo, then μx(0) exists.j

Proof. Suppose that I xnn~ι)p{x) dx = oo and det Z(x) > 0 on [0, oo).Then by the preceding theorem, det Y(x) > 0 on (0, oo) and thematrix K{x) = Y{x)Z~\x) is positive definite on (0, oo). Thus, for everyconstant vector ξ — (§;), (i = 1, 2, , w), it follows that

Hence, if #0 e (0, oo) and x e (x09 oo), it follows that

ξ*F(t)ξdt < ξ*K-\xo)ξ < oo ,o

and the theorem follows.In order to use to advantage the second-order equation (16) which

ρ(x) satisfies (as Barrett does in [2]), it must first be established thatthe coefficients in (16) are positive on (0, oo). Lemma 3.2 gives p(x) > 0on [0, oo). Thus, it remains to be shown that S^x), as defined by (17),is positive on (0, oo). In fact, it is necessary to know that a certainfamily of determinants (which includes Sλ(x)) contains only determinantswhich are nonzero on (0, co) and that certain of these (including δ^x))are positive on (0, oo). These results are given in Lemma 4.1.

Consider the matrix M{x) with 2n rows and n — 1 columns (n > 1)whose first n rows are, in order, the row vectors (u[k)u{

2

k) u^), k =0,1, "',n — 1, and whose last n rows are, in order, the row vectors(u[kMk)i ' tt£llfl), k = 0,1, , n - 1. Let j^~" denote the family ofdeterminants of order n — 1 obtained from M(x) by deleting n + 1 rowsand taking the determinant of the resulting square matrix. Note thatfor every determinant D(x) in ^ " , D(0) = 0, except the case D(x) =p(x) when D(0) ~ 1. Two subsets of j^~ are the sets {8j(x)/p(x)} and{7j(x)}, j = 1, , n — 2 (n > 2) defined as follows:

DEFINITION 4.1. Sά(x)lp(x) is a determinant of order n — 1 with firstrow (%!, %2 ^n-i) and remaining rows, in order, (^^ί^ί wi-i.i)> A; =0,1, , — j — 2, w — j , , n — 2.

DEFINITION 4.2. γy(a?) is a determinant of order n — 1 with rows, inorder, « K i uH°lltl), k = 0,1, , n - i ~ 3, n - i - 1, , n - 1.

LEMMA 4.1. If D{x) is any determinant belonging to j ^ ~ , thenD(x) Φ-0 on (0, oo). Furthermore, if D(x) is any determinant in the

set {8j(x)lp(x)} or in the set {7j(x)}, j = 1, , n — 2, then D(x) > 0 on(0, oo).

Proof. A general determinant D(x) in &~ will have the form:

D(x) =

t t j<<1)

where for some fe,0^i;^tι-l, {ίiK=ϊ are some fc of the integersbetween 0 and n — 1 and 0 ^ iλ < ί2 < < iΛ ^ n — 1; and {ίilpfcTlare some n — 1 — k of the integers between 0 and n — 1 and 0 ^ i

The method of proof of the first part of the lemma, which willnow be outlined, is the same as that of Lemmas 3.1 and 3.2 exceptfor the added generality. In fact these two lemmas are included inthe present lemma, but were established separately and in the lastsection for clarity.

Assume that D(x) has a zero on (0, oo). Then there is a solutiony(x) of (1) which is a linear combination of uλ(x)9 u2(x), •• ,un-1(x) anda minimum value x = b on (0, oo) such that y(0) = y'(Q) — — ^ ( O )— 2 / ( ? ) ~ 1 ) ( 0 ) = 0 a n d y { i χ ) { b ) = y { i 2 ) ( b ) = = y i i ] c ) ( b ) = y [ i j c + l ) ( b ) = = • • • =

yUn-tfφ) — o. Recall that yx{x) = r(x)y{n)(x). Rollers theorem can nowbe applied successively to y(x) and, as in Lemma 3.1, a contradiction tothe assumption is obtained to give the first part of the lemma.

The technique for establishing the second part of the lemma in-volves an iterated differentiation procedure which will now be described.Since, for all j , δi(0)/p(0) = 0 and 7, (0) = 0, a straightforward methodof establishing the desired result should be that of showing that foreach j the first nonvanishing derivative of 8J(X)IP(X) at x = 0 or ofΎj(x) at x — 0 is positive. This requires modification, however, sincethe assumptions on the continuity of p(x) are not sufficient to obtainall of the necessary derivatives directly.

Note the determinant forms of 8J(X)IP(X) and 7ό{x). The followingdiscussion applies equally well to members of either of the sets ofdeterminants. Differentiate the determinant D(x) successively untileither:

(i) One of the determinants arising in the differentiation processis a positive constant times p(x), or

956 ROBERT W. HUNT

(ii) One or more of the determinants arising in the differentiationprocess has the row vector {u[nl, u{

2

nl ••• ul?2ltl) in the last row and somevector other than (ulf u2 un^) in the first row (for in this case thedeterminant is zero since u{"}(x) = ( — l)np(x)Uj(x)).

If (i) occurs first or in the same step as (ii) after m differentiations,then D{h)(0) — 0 for k = 1, , m — 1 since all of the determinantsobtained belong to ^ and l5(m)(0) = kp(0) = k for some positive con-stant k, k being the number of times that determinant has appeared inthe differentiation process. Thus, D(x) > 0 on (0, oo) since D(x) Φ 0 on(0, oo) and its first nonvanishing derivative at x = 0 exists and ispositive.

If (ii) occurs first after m differentiations, the determinants of theform given in (ii) can be altered in form by replacing the last row by( — l)np(x) (ulfu2 ••• un^) and then bringing this row to the position ofthe first row and letting rows 1,2, •• ,n — 2 become rows 2,3, •••,n — 1, respectively. The latter operation requires n — 2 row inter-changes so that each of these determinants is now some member of^ (and not of the sets {S3lp} or {γ, }) multiplied by + p(x). Let x0 besome positive finite number and denote the maximum and minimumvalues of p(x) on [0, x0] by pM and pm, respectively. Then, for each ofthese determinants multiplied by p{x), replace p(x) by pm if that de-terminant is positive on (0, oo) and replace p(x) by pM if that determi-nant is negative on (0, oo). Recall that each determinant is nonzeroby the first part of the lemma. On [0, x0], let fx(x) denote the newfunction obtained from D{m)(x) by making these changes in certaindeterminants of D{m)(x). Then D{m)(x) ^ fx{x) on [0, x0] and D{m)(0) =/i(0) - 0.

Next, on [0, x0], defferentiate fx(x) successively until (i) or (ii)occurs and then repeat the entire process just described. This processcan be continued until for some minimum integer h and some integerq, a function f^]{x) (on [0, #0]) with the following properties is obtained./w(0) = kp(0) = k>0, and /^(0) - 0 for j = 0,1, , q - 1. Thenfh(%) > 0 in some right-hand neighborhood of x = 0. Finally, from theset of inequalities obtained, from the fact that each/^x) (j = 1, 2, , hif q > 0 and j = 1,2, , h — 1 if q = 0) and its appropriate derivativesare all zero for x — 0, and from the first part of the lemma, it followsthat D(x) > 0 on (0, oo).

The following generalization of a theorem of Barrett [2] gives aweaker condition for the existence of a / (O) for (1) and thus a strongerresult than Theorem 4.2 (and hence a better result than Reid's results[12] give for this case since Theorem 4.2 coincides with Reid's resultin [12]).

p(x)(I pfdx = oo, where I p is the nth Her-xo

ated integral of p(x), then μ^O) exists.

Proof. From Lemma 3.2 and Lemma 4.1, it follows that (16) haspositive coefficients on (0, oo). Since ^(0) = 1, ρ'(0) = 0, a well-knownresult for second-order, self-adjoint differential equations [2, Theorem1.3] can be applied to give the existence of a zero of p(x) on (0, oo) if

S CO

p{x)p\x)dx — oo. But, using Theorem 2.1, a zero ofp{x) on (0, oo) implies the existence of a /*i(0).

Thus, Theorem 4.3 will be proved if it can be shown that

ρ(x) ^kTp , x0 > 0, k > 0 ,

for then the hypothesis of the theorem implies that the desired prop-

S oo

p{x)p\x)dx = oo, is true.Now, p(x) = W[ultl(x),u2Λ(x), , un-1Λ(x)] and differentiation yields

ρ"(x) = γx(χ) + s^x), where 7λ(x) and S^x) are as previously defined.Then, consider

p"{x) - Sx(x) and [p»(x) - δ^x)]' = Ύ2(x) + S2(x) .Next, \{p" — Si)' — δ2]' = γ3 + δ3 is obtained, and the process can be

continued to obtain finally

» _ Slγ - 80' - 8.y δw_4)' - δ._8]' - γ%_2 + δ%_2

where 7i(ic) and δj(x), j = 1, , w — 2, are as previously defined.

Then, one more differentiation gives

[(• (((£" - SO' - δ2)' - 80' 8n_0' - δ._J' = φ)8*{x) ,

where 8*(x) is a detei^ninant of order n — 1 with first row (ul9 u2

%n_0 and remaining rows

« ί , u£l <*Λ.i) for k = 1, 2, , n - 2.

Now, by using Lemma 4.1 and the fact that γ;_2 = p(a?)8*(a?), δ*(α?)must be positive on (0, oo). Also, by the techniques of Lemma 4.1,δ*'(x) is positive on (0, oo). Thus S*(x) is an increasing function on(0, oo) and for any x0 > 0,

" P(t)δ*(t)dt 2: 8*(xo)\" p(t)dt = k Tpy k > 0 .x0 J x0 XQ

Now, using Lemma 4.1, the expression

XQ

958 ROBERT W. HUNT

can be integrated n — 1 times and the inequality preserved. This givesp(x) ^ k IX

XQ p, the result needed to complete the proof of the theorem.Note that for 0 < x0 < xx ^ x < oo,

P(x)dx,VU — 1)1

and thus Theorem 4.2 follows from Theorem 4.3. Hence, as remarkedpreviously, Theorem 4.3 gives the better condition for the existence ofμx{b) for any b ε [0, oo).

5. Conditions for the existence of ^(0) Consider the matrix H(x)= —Z(x)Y-\x) defined on (0,^(0)). Note that H(x) is then equal to— K~\x), where K{x) is the matrix used in Theorem 4.1 and satisfying(18). Then H(x) is symmetric on (0, ^(0)) and satisfies

(19) H'{x) = F{x) + H{x)E{x)H(x) ,

as in [1] and [11],Note that H(x) is defined and symmetric on (0, oo) if there does

not exist an i(O). Then consider the following lemma.

LEMMA 5.1. If ^(0) does not exist for (1), then all the eigenvaluesof H(x) are nondecreasing on (0, co).

Proof. From the properties of the matrices E{x) and F(x) asdiscussed in § 1 and from (19), it follows that H\x) ^ 0 on (0, oo) ifthe inequality is used to imply positive semi-definiteness of H'(x)« ThenH(x2) — H(x^) is positive semi-definite if 0 < x1 < x2 < °°, i.e., H(x2) >H{x^). Then, from classical extremizing properties of eigenvalues, theresult of the lemma follows.

In the present notation, / (O) denotes the^ first zero of p(x) on(0, oo). In addition, let μ.2(0) be the second distinct zero of ^(0) on(0, oo), μ3(0) the third distinct zero of p(x) on (0, oo), and so forth.

THEOREM 5.1. // ^(0), μ2(0), μ3(0), , μn(0), /W0) all exist (i.e.,if P(χ) has n + 1 distinct zeros on (0, co)), then i(O) exists.

Proof. Suppose i(O) does not exist so that Lemma 5.1 applies.Also, H(x) = — Z{x)Y~\x) is defined on (0, oo) and has singularities atthe ^(0) so that det H(ft(0)) = 0, i = 1, 2, , n + 1. But det H(x)is equal to the product of the eigenvalues of H(x) and thus, by Lemma5.1, det H(x) can vanish at most n times on (0, oo) unless det H(x) = 0on a subinterval of (0, oo). Assume the latter is true. Then det Z(x)= ρ(χ) = 0 on a subinterval of (0, oo). But ^(0) = 1 and ρ{x) satisfies


the linear, second order, self-adjoint differential equation (16) whichimplies that p(x) cannot be identically zero on a sub-interval of (0, oo)without being identically zero on the entire half line. This contradic-tion completes the proof of the theorem.

THEOREM 5.2. // μ^O) exists and \ dxlr(x) — oo, then ηλ{0) exists.

Proof. Assume that μ^O) does not exist. Then, using Lemma 5.1and Theorem 5.1, it follows that the maximum eigenvalue of H(μλ{ϋ))— 0 and that there is a value x — x0 on (/A(0), oo) such that the maxi-mum eigenvalue of H(x) is positive on [x0, oo). Furthermore, there isa value x = x1 xQ such that H(x) is nonsingular on [x19 oo). Then,from (19), H-^H^H^x) ^ E(x) on [xl9 oo). Also, if ξ is any con-stant column vector of unit length, then on [x19 oo),

(20) ξ*H~\x)ξ ^ max eigenvalue of H-\xλ)

— min eigenvalue i E(t)dt .

Now, noting the form of |*( 1 E(t)dt)ξ (as shown in the proof ofVJo /

Theorem 4.1), where ξ is an arbitrary, nonzero, wth-order, constantvector, [°°dxlr(x) = oo implies that W(~δ7(ί)dί)f = oo. But Reid [12]has proved a result which, in the terminology of the present problem,

E(t)dt)ξ = oo for arbitrary, nonzero, constant vec-0 / Γx

tors, then the minimum eigenvalue of\ E(t)dt approaches oo as x ap-Jo

proaches oo. Then, it follows from (20) that there is a values x = x2

on (xl9 oo) such that H~\x) and H(x) are negative definite on [x2, oo).This contradicts the fact that the maximum eigenvalue of H(x) ispositive on [x0, oo). This contradiction gives the result of the theorem.

Theorem 5.2 can be obtained from Reid's results in [12]. The proofgiven here is entirely different, however, from that in [12].

p(x)(I pfdx = oo and \ dx/r(x) = co,x0 J

^(0) exists (n > 1).

then

Proof. This result follows immediately by combining Theorems 4.3and 5.2.

6 Conditions for the existence of μλ(0) and (0) for (4). Fromthe general results of Reid's [12], two theorems can be obtained andthen used to prove two comparison theorems pertinent to the present dis-

960 ROBERT W. HUNT

cussion. Then a comparison of (1) and (4) can be give conditions forthe existence of / (O) and %(0) for (4), a more general self-ad jointequation then (1). For (4), ^(0) is defined as in Definition 2.1 and (10).However, for (4), the definition of / (O) is obtained by altering Defini-tion 2.2 by replacing (11) by

(11') y(0) = y'(0) = = y (-1 }(0) - 0 - ^(6) - y2(b) = . . . = yn(b) ,

where yx(x) = r(x)y{n)(x) as before and

Vi(χ) = (rMyln)r-1} + Σi-iy+Kr^xW-'y-t-v, ΐ - 2,3,. ., n

Reid's results are stated in the notation of the present discussion.

THEOREM 6.1. (Reid [12]) A necessary and sufficient condition forthe nonexistence of %(0) on (0, c] for (1) is that

(21) IJy; 0, δ] - ( W ) 0 / ( w ) ) 2 - P(x)y2]dx > 0Jo

/or all values ofb on (0, c] and all functions y(x) such that y(x) e C^fO, δ],y(x) is absolutely continuous on [0, δ], (^/(%))2 ^ s integrable on [0, δ], α^d?/(#) feαs nth-order zeros at x = 0 α^d α? = 6. Λiso, i/ %(0) exists fora solution y(x) of (1), ίfeew /Jy; 0, %(0)] = 0.

For the equation (4), the statement analogous to (21) is

(22) I2[y; 0, δ] - \\r(x)(y^y - ^{x^y^dx > 0 ,JO i=0

where τ/(#) is the same as in the statement of Theorem 6.1.

THEOREM 6.2. (Reid [12]) A necessary and sufficient condition forthe nonexistence of βλ(O) on (0, c] for (1) is that (21) (or (22) if (4) isused) hold for all values of b on (0, c) and all functions y(x) such thaty(x) G C*-1^, δ], y(x) is absolutely continuous on [0, δ], (y{n)f is inte-grable on [0, δ] and y(x) has an nth-order zero at x — 0. Also, if μλ(Q)exists for a solution y(x) of (1), then I^y; 0, /A(0)] = 0.

Theorem 6.1 can now be utilized to obtain the following results.

THEOREM 6.3. // %(0) exists for (1) at x = xlf r(x) ^ rn(x), andp(x) < ro(x), then ^(0) exists for (4) at x — x2, say, and 0 < x2 a?1#

THEOREM 6.4. // μλ(0) exists for (1) at x = x19 r(x) ^ rw(a?),p(x) ^ ro(aj), ίΛβ^ ^(0) exists for (4) αί x = x2, say, and 0 < x2 ajle

Now, take r(#) = r%(αj) and p(x) = ro(x) on [0, oo); and then thefollowing results are obtained by combining the results of § 4 and 5 and


those of the present section.

S oo χn

ro(x)(I rofdx = co, then μjb) exists for (4) forXQ

each b e [0, oo).

S oo

dx/rn(x) = oo, then %(0)exists for (4).

S X f oo

ro(x)(IrQydx = oo and \ dx/rn(x) = oo, then,XQ J

i(O) exists for (4).Theorem 6.5 and Corollary 6.1 give stronger results than the cor-

responding results of Reid's [12] since the latter have the condition1 x^^-Ur^x^dx = oo while the former have the weaker condition

S X

ro(x)(Iroydx. (See the proof of Theorem 4.3 for a comparison ofthese conditions.) Theorem 6.6 can be obtained from Reid's work [12].

BIBLIOGRAPHY

1. J. H. Barrett, A Prύfer transformation for matrix differential equations, Proc. Amer.Math. Soc, 8 (1957), 510-518.2. , Disconjugacy of a self-adjoint differential equation of the fourth order,Pacific J. Math, 11 (1961), 25-37.3. , Systems-disconjugacy of a fourth-order differential equation, Technical Sum-mary Report No. 135, Math. Research Center, U. S. Army (1960) 12 pp. Proc. Amer. Math.Soc, 12 (1961), 205-213.4. , Two-point boundary value problems and comparison theorems for fourth-orderself-adjoint differential equations and second-order matrix differential systems, CanadianJ. of Math., 13 (1961), 625-638.5. E. Bodewig, Matrix Calculus, North-Holland Publishing Co., Amsterdam, 1956.6. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGrawHill, New York, 1955.7. R. Courant and D. Hubert, Methods of Mathematical Physics, Vol. I, Interscience, NewYork, 1953.8. H. C. Howard, Oscillation criteria for fourth-order linear differential equations, Trans,Amer. Math. Soc, 96 (1960), 296-311.9. H. Kaufman and R. L. Sternberg, A two-point boundary problem for ordinary self-adjoint differential equations of even order, Duke Math J., 20 (1953), 527-531.10. Walter Leighton and Z. Nehari, On the oscillation of solutions of self-adjoint lineardifferential equations of the fourth order, Trans. Amer. Math. Soc, 89 (1958), 325-377.11. W. T. Reid, A matrix differential equation of the Riccati type, Amer. J. of Math.,6 8 (1946), 237-246.12. , Oscillation criteria for self-adjoint differential systems, Trans. Amer. Math.Soc, 1O1 (1961), 91-106.13. H. M. Sternberg and R. L. Sternberg, A two-point boundary problem for ordinary self-adjoint differential equations of fourth order, Canadian J. Math., 6 (1954), 416-419.14. R. L. Sternberg, Variational methods and non-oscillation theorems for systems ofdifferential equations, Duke Math. J., 19 (1952), 311-322.

SOUTHERN ILLINOIS UNIVERSITY

A REMARK ON THE NIJENHUIS TENSOR

EDWARD T. KOBAYASHI

The vanishing of the Nijenhuis tensor of the almost complex structureis known to give the integrability of the almost complex structure [3, 7].In order to generalize this fact, we consider a vector 1-form λ o n amanifold M [4], whose Jordan canonical form at all points on M is equalto a fixed matrix μ. Following the idea of E. Cartan, we say that sucha vector 1-form is O-deformable [2]. The frames z at x such that z~~xhxz =μ define a subbundle of the frame bundle over M, as x runs throughM, and the subbundle is called a G-structure defined by h [1]. We findthat for a certain type of O-deformable h, the vanishing of the Nijenhuistensor of h is sufficient for the G-structure to be integrable (Theorem,§2). In § 5 we give an example of a O-deformable derogatory nilpotentvector 1-form, whose Nijenhuis tensor vanishes, but whose G-structureis not integrable.

1. Vector forms and distributions As usual, we begin by stating,that all the objects we encounter in this paper are assumed to be C°°.

Let M be a manifold, Tx the tangent space at point x of M, T thetangent bundle over M, T{p) the vector bundle of tangential covariantp-vectors of M. A vector p-form is a cross-section of T®T{V). Thecollection of all vector p-forms over M is denoted by Ψp. We noticethat a vector 1-form is nothing but a law that assigns a linear transforma-tion to each tangent space Tx at point x of M.

We list some definitions and lemmas of the theory of vector forms[4], which we use in the sequel.

If PeΨp,Qe Ψqy then P A Q e Ψp+q^ is defined by

(1) (PA Q)(ult •• ,up +,_1)

where a runs through all the permutations of (1, 2, •••,# + (/ — 1), and\a\ denotes the signature of the permutation a.

If & is a vector 1-f orm and P is a vector p-ΐorm, we write hP in-stead of h 7\ p. In particular if p = h, we write h A h as h\ In general,h A h A h is written as hk, and this agrees with the usual notation,

k t

Received November 13, 1961. This research was supported in part by National ScienceFoundation grant G 14736. The author wishes to thank Professor H. C. Wang for first sug-gesting this problem and for the subsequent discussion we had on this paper. The authoris also obliged to the referee whose comments helped to correct and clarify some argumentsin this paper.

963

964 EDWARD T. KOBAYASHI

when we consider feasa linear transformation of the tangent space ateach point of the manifold M.

Let ft and k be two vector 1-forms. The bracket [ft, k] of ft andk is a vector 2-form defined by

(2) [h, fe](u, v) = [fiu, fei>] + [feu, ftα>] — fc[ftu, v] - h[ku, v]

—k[u, hv] — h[u, kv] + kh[u, v]

where u and v are vector fields over M. If ft = k9 we obtain the tensor[ft, ft], generally known as the Nijenhuis tensor:

(3) JL [h9 h](u, v) - [hu, hv] - h[hu, v] - h[u, hv] + h*[u, v] .

If h, k and I are vector 1-forms, using (2), we can obtain

(4) [hi, k) + [h, kl] -[h,k]7\l = h[l, k] + k[l, h]

(cf. (6.7) [4]).

LEMMA 1.1. Let h be a vector 1-form, then

(5) [hk, hι] = i - Σ ha{{[h, h] 7\ hb) 7\ h° - [h, h] A hb+°} .9 a+b + + k + l2

Proof. By replacing ft, & and I by ft, ft and ftfc in (4), we obtain

(6) [ft\ ft] - ft[ft&'\ ft] + -ί[ft, ft] A ft*"1 ,

which gives us

(7) \h\ Λ] = — Σ Λ'"1^, Λ] A ^ - { .2 <=i

Again, replacing ft, k and £ in (4) by ft\ ft and ft1""1, we obtain

(8) [hk+ι~\ ft] + [ft\ ft'] - [ft&, ft] A ft'"1 = ft*[ft*-\ ft] + ft[ft*~\ hk] .

Using (7) and (8) yields

(9) [ft*, hι] = h[hk, ft'"1]

+ - Σ f t ^ m λ] A h"-*) A ft^1 - [ft, ft] A ft*-^"1} ,2 •=»!

and repeating the reduction we obtain (5).

LEMMA 1.2. Let h be a vector 1-form on M, whose rank is constant

A REMARK ON THE NIJENHUIS TENSOR 965

in a neighbourhood of each point x of M. If [h, h] = 0, the distribu-tion x-+hxTx is completely integrable.

Proof. By Frobenius' theorem we have to show that the bracketof any two vector fields of the form hu, hv belongs to the distribution.This follows from [h, h] = 0 and (3):

[hu, hv] = h[hu, v] + h\u, hv] — h\u, v] .

We recall that a necessary and sufficient condition for a distributionto be completely integrable can be given as follows:

Let θ be an r-dimensional distribution x —> θ(x) on an m-dimensionalmanifold M. For each x0 e M, let U be a neighbourhood of x0 andLlf •••, Lr be vector fields on U such that (L^x, , (Lr)x span θ(x) foreach xe U. Then θ is completely integrable if and only if for eachx0 e M, there exist m — r independent functions ψ1, , ψm~r defined ona neighbourhood V c U of xQ such that

Liψ3 = 0, for 1 S i ^ r, 1 ^ i ^ m - r on V .

Using this it is easy to prove,

LEMMA 1.3. // θu , θg are completely integrable distributions ofdimensions rlf , rg on M, such that

θx{x) + Θ2(x) + + θg{x) = Tx (direct sum)

for each xe M, then for each point x0e M, there exists a coordinateneighbourhood U of x0 with coordinate functions x1, , xm such thatfor each j

gives an integral manifold of θ5 contained in U.

2. The integrability of a O'deformable vector 1-foriru Let h be avector 1-form, defined on M, whose characteristic polynomial has con-stant coefficients on M. Let the decomposition of the characteristicpolynomial be

where ^(λ), i = 1, , g are polynomials in λ, irreducible over the reals,and (Pi(X), P,(λ)) = 1, if i Φ j . It is easy to verify [5, pp 130-132], thatwe can get polynomials β^λ), e2(λ), , eg(\) in λ, with constant coef-ficients, such that ΣiUMh) = /, {e^K)}2 = e{(h), e^'e^k) = 0 f or i Φ j ,and


et(he)T9 - K e Tx I {Vi{K)Ymx = 0} .

Let θi denote the distribution x--> e{{hx)Tx. If we assume [h, h] = 0,then by Lemma 1.1, because e^h) is a polynomial in h with constantcoefficients, we see that [e^k), e^h)] = 0. Hence, by Lemma 1.2, θi iscompletely integrable.

DEFINITION. A vector 1-form h on M is said to be 0-deformable, iffor all x e AT, the Jordan canonical form of Λβ is equal to a fixed matrixμ [2].

Note that a 0-deformable vector 1-form has a characteristic poly-nomial with constant coefficients.

A frame at xeM is an isomorphism 2 from Rm onto 7 , where mis the dimension of M. For a 0-deformable vector 1-form h, the framesz at x such that 2~3/&xz = μ define a subbundle if of the frame bundleover M, as x runs through Λf. H is called the G-structure defined by

M i ] .

DEFINITION. A G-structure H defined by h is said to be integrable,if for each point x of M there exists a coordinate neighbourhood U ofx with a coordinate system {x1, ,xm} such that the frame {(d/dx1),.,, ,(0/0$"%,} belongs to the subbundle H for all xr e U. We shall say thatthese coordinate functions are associated with the integrable G-structureH.

Clearly, H is integrable if and only if, for each point x of M, wecan find a local coordinate system around x, in which the coordinateexpression of h is μ.

We are interested in finding a sufficient condition for a G-structuredefined by a 0-deformable vector 1-form h to be integrable. We nowassume [h, h] — 0. By the argument above we know that the distribu-tions θi associated to the irreducible factors ^(λ) are all completelyintegrable, so by Lemma 1.3, for each point x0 of M there is a coordi-nate system {x1, , xm) on a neighbourhood U of x0, and the integralmanifolds of θt contained in U are given by coordinate slices.

In U take a point given by coordinates (f1, , ξm). For each i, letx1 = ξ1, , ί* -1 = £r*-i, xr*+1 = |r*+1, , xm = £w give an integral mani-fold Mι of #£ in Z7, where r = mx + m2 + + m i and m{ = dimensionof θit Consider the restriction ht of h on Mi. Notice that we can viewhi as a vector 1-form on an open set of Mif depending on m — miparameters x1, , af*-1, xrί+1, , xm in such the way that hi is C°° withrespect to the coordinates on M{ and the parameters together. Thecharacteristic polynomial of hi is {Pi(λ)}aί and the minimum polynomialof hi is {Pi(X)}n, where HUi{Pi(^)}Vi is the minimum polynomial of h; hiis a 0-deformable vector 1-form on Mif and [hif hi] = 0. If for each i,

the G -structure defined by h{ on M{ is integrable, and if coordinatefunctions yrt-1+1, , yr% associated to the integrable G -structure aroundthe point (# r*-1 + 1, , xrή = (ξrί-1+1, , ξrί) are dependent on coordinatesχu-i+1

9 . . .f χrt and on parameters x1, , xr*-i, xri+1, , xm jointly in aC°°-manner, then we can replace {x1, « ,#m} in a neighbourhood of thepoint (a?1, , xm) = (|\ , ξm) by a new coordinate system {y1, , #w},so that h takes the matrix form μ, i.e. if is integrable.

Hence we consider the case where h has characteristic polynomial{p(X)}d and minimum polynomial {p(λ)}Ό, where j>(λ) is irreducible overthe reals, and suppose that h jointly depends on the coordinates of Mand some parameters in a C"-manner. We have the following results:

Case I. degp(λ) = 1.( i ) If v = 1, then h is a constant multiple of the identity vector

1-form / on M, hence the G-structure is integrable.(ii) If v = d = m, consider the nilpotent part n of h. n is a poly-

nomial in h with constant coefficients on M, so from [h, h] = 0, we get[n, n] = 0, by Lemma 1.1. Moreover nm — 0 but nι Φ 0 for I < m, forall points of M. In § 3 we prove a proposition which shows that theG-structure defined by n (which is the same as that defined by h) isintegrable, and that the associated coordinate functions depend on theparameters of h and on the point in M jointly in a C°°-manner.

Case II. deg p(λ) = 2. In § 4 we shall show that the semi-simplepart s of h gives rise to a complex manifold structure M in this case,and that for the G-structure given by h which is induced from h on M,(i) and (ii) of Case I has a straightforward parallel on M hence comingback to the real manifold, we have: if v = 1, or v = d = m/2, then theG-structure defined by h is integrable, and the associated coordinate func-tions are C°° with respect to the coordinates on M and the parametersjointly.

By the preceding arguments and the results in § 3 and 4, we canconclude the following:

THEOREM. Let h be a 0-deformable vector l-form on a manifold M,with characteristic polynomial

where Pi(X) are polynomials in λ, irreducible over the reals, and (Pi(X),

— 1 for i =£ 3> and the minimum polynomial

Suppose for each i, vt = 1 or d{. Then the G-structure defined by h isintegrable if [h, h] = 0.

REMARK. If v{ — 1 for all i, we say that h is semi-simple. If vt = cίfor all i, we say that /& is nonderogatory, and otherwise derogatory [6,P. 21].

3* The integrability of a nonderogatory nilpotent vector Inform •

PROPOSITION. Let hbe a nilpotent vector 1-form on an m-dimensionalmanifold M, and suppose hm = 0 but hι Φ 0 for I < m, /or αϊi points onM. Then [h, h] = 0 implies that the G-structure defined by h is integrable.Moreover, if h depends on some parameters and is C°° with respect tothe local coordinates x\ " ,xm on M and the parameters jointly, thenthe local coordinates y1, , ym associated to the integrable G-structureare C°° with respect to x1, •• ,xm and the parameters jointly.

Proof. (1) Let m = 2. Denoting the tangent space at x e M by Tx,we have a one dimensional distribution given by x-^hxTx. For each pointx0 of M we can find a neighbourhood U of x0 and a coordinate system{x1, x2} on U, such that x2 — ξ2 is an integral manifold of this distribu-tion in U. Let h take the matrix form in this coordinate system

β* A

βi3 being functions of x1, x2. As dfdx1 at xe U spans hxTx, we haveβ*. = /?22 = 0, and as A restricted to integral manifold x2 = f2 is givenby βiu and as fe2 = 0, we have βn = 0. We claim, that we can choosea new coordinate system {y1, y2} such that in this new coordinate systemh takes the matrix form

0 1

0 0

In fact, let the vector fields d/dx1 and djdx2 be denoted by X± and X2,and choose new vector fields Y± and Y2 by

where α2 and α:0 are to be determined so that /&F2= Y1 and [F l f y j = 0.Let then π\ π2 be the 1-forms dual to Yly Y2; we have dπ1 = 0, dπ2 = 0,so that y1, y2 can be determined from dy1 = π1, cί?/2 = ττ2. To prove thatYi and F 2 can be found we observe that the condition hY2 = Yx leadsto


a! = β12

and that the condition [Ylf Y2] — 0 leads to

which is a first order linear differential equation for α0:

"dx1 ° ° W V dx2 1

&! is clearly C°° with respect to x1, x2 and the parameters. aQ is obtainedas a solution of the above differential equation, so a0 depends on x2 andthe parameters in a C°° manner. By differentiating this differentialequation repeatedly, we see that a0 is C°° with respect to x1, x2 and theparameters. Hence π1 and π2 are C°° with respect to x1, x2 and the parame-ters, and finally y1 and y2 are C°° with respect to x1, x2 and the parameters.

(2) We assume that our proposition is true for (m — l)-dimensionalmanifolds and proceed to prove it for an m-dimensional manifold (m 3).

Because [h, h] = 0, we know that the distribution x—>hxTx, given bythe image of h at each point x of M is integrable; hence, locally, thereexists a coordinate system {x1, , xm) such that

( i ) χm = ξm gives the integral manifolds of this distribution, and(ii) in this coordinate system h takes the matrix form

/ Am \/ \

H

•0

•

β,

0

n—1 m

/

TVe further claim that x\ •••, xm~\ xm can be chosen so that(iii) H takes the form

/0 1 0 0\

• 0 1 0•

0 1

\0 0/

In fact, if H is not in the form (2) already, we view the restriction hx

of h to an integral manifold xm = ξm as a vector 1-form on an open setV of Rm~x, depending on parameter xm, and consider H to be the matrixform of hλ with respect to the coordinate system {x1, •• ,#m-1}. Fromthe inductive assumption, there are coordinate functions z1, , zm~1 onan open set F x c F depending on x1, •• ,#m~ 1 and xm in a C°°-manner,such that hx has matrix form (2) with respect to the coordinate system

{z1, •• ,2m~1}. Now, if we take {z1, • • ,2m~1, xm} as the local coordinatesystem on M, then (iii) will be satisfied.

So let us suppose that we are in a coordinate system where (i) (ii>and (iii) are satisfied. For simplicity we write A, A, , A*-i insteadof Am, Am, , A»-im. Note that A»-i Φ 0. We want to prove that wecan find a new coordinate system {y1, , ym} such that in this coordi-nate system h takes the matrix form (1), H being of the form (2) andβλ = β2 = . . . = βm_2 = o, A,-! = 1. In order to do this, as in the casem = 2, we find vector fields Yu ,Ym satisfying hYt= Yi_i (i = 2, , m),hY1 = 0 and [Γi, Γy] = 0 for all i,j; let the dual of Yl9 •••, F m beπ1, , πm and obtain y1, , ym from dy1 = π1, , cZτ/m = ττm. If wedenote by Xu , Xm the vector fields β/βα?1, , d/dx™ and set

(3)

m _ ! = αTi-Xi + a2X2

[Ym - aQXx + (a, -

where αm_j = /3TO_i, then the problem reduces to finding the a's so that[Yif Yj] = 0 are satisfied for all i,j.

First we shall obtain all the relations on the derivatives of A, " vA»-i imposed by the condition [h, h] = 0. We see that

[h, h](Xif Xj) = 0

gives us no relations for i, j <=; m — 1, but

\[h9 h](Xi9 Xm) = [X,_u AXx + + Aa-Λ-l]

- h[X{, β1X1+" + β.-rX.-r]

from which we obtain

(4) Xt-J3s-x = ^ ^ ί i , j ^ m - l

and

(5) X^m_ x = 0 t^m-2.

To make this relation clear, we write this result in Table 1.

0 = Xβm-2 = Xβm-1

χ& =

TABLE 1

Now let us examine [ Y , F, ] = 0 for i < j ^ m — 1. We see thatthis is equivalent to the set of equations (6),

r frXi + am-i+1X2 + + α:m_1Xΐ)αw_1 = 0

(6)

- +i = 0

-iXi + am_i+1X2 + + a^X^n-j+i-!

_1 = 0

-i-Xi + αm_ ί+1X2 + +

- (α^yjζ + am_j+1X2 + + cc^Xjyx^ = 0

0 - Xλam-x0 = X1am_2 = X2am.x

where i < i ^ m — 1. Using Xi«:w-i = Xβm-i = 0 from Table 1, we seethat (6) is equivalent to the following Table 2.

(a)

— •Λ-m-2CXm-2 — ^-w-i α m-l,

0 = Xλa, = X2a, = . . . =

0 = I A = X2α3 = =

X-βL-L = X2Oί2 = =

(b)

TABLE 2

Next consider [Yif Yn] = 0, i ^ m — 1. This is equivalent to thefollowing (7a, b, c),


(α«-.*Xi + ocm_i+1X2 + + αm_1X i)(«»-2 - &.-0 = 0

(7a)

( . - ί X x + αm- i + 1X3 + + α.^X i )(α 1 - &) = 0

-iXi + αm_i+1X2 + + αro_1Xi)(«i-i - A-i)

( α . _ - /3ro_2)Xm_1 + X J α ^ = 0

( α . - Λ + «m-ί+1X2 + + α._1Z4χα1 - A)- {a0X, + (a, - A)X2 + + (α.-, - /3»-2)X»-i + Xm}am_i+1 = 0

(7b) j

(ff,.iZ, + αTO-ί+1X2 + + α ,

- {ctoXi + fa - A)X2 + + (α -, - /5w-2)XTO-i + Xm}«m-i = 0

where i ^ m — 1.Because of Table 1, we see that (7a) is equivalent to part (a) of

Table 2. Using part (a) of Table 2, we see that (7b) reduces to a simplersystem (7b'),

(7b') {

(am_i+1X2 + + a^XfYflc - A)

-i+1 = 0

Using Table 1 again, we can show that (7b') is equivalent to part (b)of Table 2 plus the following equations which are obtained from (7b'>by letting i = m — 1:

»_1X._1)(α»-1 - /3m_2) - {(α»_, - βm-,)Xm-x + Xm}um-ι = 0

(α2X2 + + α._1X»_1)(α1 - ft)

- {(αi - A)X2 + + (α»_, - β.-JX.-i + X m R = 0

Using Table 1 and part (b) of Table 2, these equations can be writtenas (8),

(8) (pc-rYX.-! a™-k ~ ^"-* + (α.-0'X.-i g - t + 1 ~ / 9 m- f t + 1

+ + (αm_ i+1)1X.-χ α «- ' ~ β*~> - Xmam-M = 0 ," A; = 2, , m - 1 .

1 For simplicity we write (am-i-j)2Xm-i(am-k+j—βm-k+jl(Xm-i-j), 1 ^ j ^ k - 2, foram-i-jXm-i(am-k+j - βm-k+j) - (Xm-i(Xm-i-j)Xm-i((Xm-k+j — βm-k+j), although at somepoint αm_i_^ might vanish.


We can now obtain tfm_2, α»-8> > ai succesively by integrating (8)with respect to x*-1; in fact, start from k = 2, and integrate to get αm_2,then use this α:m_2 in (8) for k — 3 and integrate to get αm_3, in general

We still have to show that αw_2, αw_3, , oil thus obtained satisfyTable 2. For simplicity let us write (8) in the form

Then (9) becomes

To show that the α's do satisfy Table 2, it suffices to show (10^),

(10,) Xm_,(αm_, - βm_k) = Xm_g + 1(αm_,+ 1 - /3m_fc+1)

for fc, g = 2, , m — 1. We shall prove (10fc) inductively. For k = 2it is easy to check. Suppose (102), "^(lOfe-!) are true; using this as-sumption, we differentiate (9fc) and get (11),

(11) Xm-q{am-u ~ /?.-) = αm_,

q+l^m-k + 2 -Γ

If q > 2, then Xm_gαm_2 = 0, so (11) gives us (10*). If q = 2, we observefirst that differentiating (8fc+1) with respect to xm~x gives us (12),

(12) (Xi-i(α*- f c + i - ^ - i f c + O ^ - ! - (««-*+! - /β»-fc+1)-Xi-1α.-i

Using (12) and Xm_ 2(αm_ 2 - βm_2) = 0 in (11) for g = 2, we obtain

X m _ 2 (α m _, - βm_k) = α m

which completes the proof (10fc).Finally to obtain α0, we examine (7c), and find that the same type

of argument employed to obtain (8) enables us to show that (7c) isequivalent to


(Xxa0 = Xm^(am^2 - /3m_2)

(13) Xm-2a0 =

m^ = 0 .

Using the first m — 2 equations of (13) in the last one, gives us (8k) fork = m, where we agree that /30 — 0. Hence we obtain a0 from (9m).To check that the first m —2 equations in (13) are satisfied by this a0,we check (10fc) for Jc = m. The same argument in (11) holds for k = m,and it is even simpler than before, because in this case the first termin the integrand vanishes.

If h depends on x1, « , # m and some parameters jointly in a C°°-manner, then it is clear that αm_2, •• fa1,a0 obtained above depend onx1, •• , # m and the parameters in a C°°-manner, hence we can claim thesame for y1, , ym.

4. The complex case. For Case II in § 2, where deg p(X) — 2, wehave dim M = m — 2n. Let the roots of p(λ) — 0 be σ ± iτ (τ Φ 0).Because the semi-simple part s of h is a polynomial in /& with constantcoefficients, from [h, h] = 0, via Lemma 1.1, we get [s, s] = 0. The vector1-form J8 defined by

Js = i-(β - σJ)

satisfies λ2 + 1 = 0, because s satisfies p(X) = 0. So we have an almostcomplex structure Js on M, and as [ Js, Js] = 0 (because [s, s] = 0), thisalmost complex structure is integrable [7]. Hence we can introduce anew real local coordinate system {x1, , xm) such that zk — x21*"1 + ίx2k

(k — 1, *',n) gives a local complex coordinate system, with which Mbecomes the underlying C°°-manifold of complex manifold M. As h isC°° with respect to the coordinates on M and the parameters jointly, sois the almost complex structure Js. Hence the new coordinate functionsx1, « ,xm are also C°° with respect to the coordinates on M and theparameters jointly [7].2 h is now C°° with respect to x\ , xm and theparameters jointly. The vector 1-forms on M induce vector 1-forms onM in a natural way. The vector 1-form s on M induced by s is equalto pϊ, where p — σ + iτ and I is the identity vector 1-f rom on M. Weshall show that polynomials in h with constant coefficients induce holo-morphic vector 1-forms on M. In particular, the nilpotent part n of hinduces the nilpotent holomorphic vector 1-form n on M.

2 The author wishes to thank Professor L. Nirenberg for communicating the proof ofthis fact to him. The dependence on parameters is stated without proof in [7].

Let To and T£p) be the vector bundles over M, which are obtainedΐ y complexifying the tangent space Tx and the space of tangentialcovariant p-vectors Γβ

ίp> respectively at each point x of M. Then anyp-form P on M, i.e. any cross-section of Γ ® T ( p ) , extends in a naturalway to a cross-section Po of Ta®Tp\ If k and I are two vector 1-forms on M, then kOf l0 and [kf l]0 are defined. If we define the bracketof two cross-sections of To in a natural way, and if we define [k0, l0]by (2) of § 1, where we replace h, k by k0, la acd u, v by cross-sectionsof To, then we have [k, l]σ = [&σ, Z ].

Denote 8/0s<, θ/θz* by Z i f Z« for i = 1, • •, n. (ZJ,, , (Zn)x, {Zx)x,• , (ZJ^ span the complexification of Tx. (Z1)xf , (Zn)x span the eigen-spaces of eigenvalue p. This eigenspace can be identified with the tangentspace of M at x. {Z^)x, , (Zn)x span the eigenspace of (so)x of eigen-value p. If k is a polynomial in h with constant coefficients, by Lemma1.1 we have [s, k] = 0, and hence [sσ, fcσ] = [s, &]σ = 0. On the otherhand we have

[80, ko](Zi9 Zj) = {ρ- s)[Zif k0Zj] + (p - s)[k0Zif Zd] .

s0 and k0 are polynomials in hσ with constant coefficients, so s0 and k0

commute; hence k0 leave the eigenspaces of s0 invariant, so using thecoordinate expression for k0, the equation above can be written as

[s0, ko\Z{, Z3) = <β-p)± {{Zlka) )Zk + (ZMca)M)k=l IC3

from which we get

(ko)kί is the matrix form of k on M (induced by k) with respect to thecoordinate system {z1, •••,2*}, and (1) expresses the fact that k is holo-morphic.3

( i ) If v — 1 in Case II of § 2, then h induced by h on M, is equalto s = pϊ. So in the real coordinate system {x1, , xm} h takes thematrix form

A \A 0

0\

where

τ

τ iσ3 The author is indebted to Professor H. C. Wang for this proof.

so that G-structure is integrable.(ii) If v = d — n in Case II of §2, then n satisfies nn = 0 but

nι Φ 0 for I < n for all points on iίί. As ίϊ is holomorphic, it ismeaningful to define the Nijenhuis tensor [n, n] of n, using (3) of § 1as the defining formula, where u, v should be holomorphic vector fieldson M. As [nσ, n0] = [n, n]0 = 0, we have [n, n] = 0.

Now following the method in § 3, it is easy to see that we have acomplex version of the Proposition in § 3, i.e.

"Let & be a holomorphic nilpotent vector 1-form on an ^-dimensionalcomplex manifold, and suppose kn = 0 but kι = 0 for I < n, for all points.Then [&, k] = 0 implies that the G-structure defined by k is integrable.Moreover, if k depends on some complex [real] parameters and is holo-morphic [C°°] with respect to the local coordinates z1, •••,£" [the realcoordinates a?1, , xm, where zk = x2k~τ + ix2k] and the parameters jointly,then the local coordinates w1, , wn associated to the integrable G~structure [the real coordinates 2/1, , 2/m obtained from wk = T/2^1 + ίy2k]are holomorphic [C°°] with respect to z1, , zn\xx, , α?w] and the para-meters jointly.7'

Using this complex version, for each point of M, we have a neigh-bourhood with a local complex coordinate system w1, , wn, with respectto which h — s + n takes the matrix form

0

Passing back to the real coordinate system {yι,h takes the matrix form

IABAB 0

ABA

where

- τ σ

The G-structure defined by h is thus integrable. The associated localcoordinates y1, *',ym are C°°-functions of the coordinates of M and theparameters jointly.

5. An example/ Let M be the euclidean space of dimension 4, and4 The author is indebted to Professor H. C. Wang for this example.


suppose x9 y, z, t are the coordinates. Let

Xx = d/dx, X2 = d/θy, XB = d/dz, X, = (θlθt) + (1 + z)(djdx) ,

and define h by hXx — X2, hX{ = 0 for i = 2, 3, 4. It is easy to checkthat

( i ) h* = 0,(ii) [h,h] = 0,

and (iii) [Xs, X4] = X, .Now, if the G-structure defined by h would be integrable, so would thedistributions intrinsically given by h. However, (iii) shows that the dis-tribution given by the kernel of h at each point of M is not integrable,hence we conclude that the G-structure is not integrable.

REFERENCES

1. D. Bernard, Sur la geometrie differentielles des G-structures, Ann. Inst. Fourier,Grenoble, 10 (1960), 153-273.2. E. Cartan, Sur le probleme general de la deformation, C. R. Congres Strasbourg, (1920),397-406 (Oeuvres Completes III, vol. 1).3. B. Eckmann et A. Frolicher, Sur Γintegrabilite des structures presque complexes, C. R.Paris, 232 (1951), 2284-2286.4. A. Frolicher and A. Nijenhuis, Theory of vector-valued differential forms I, Proc. Kon.Ned. Ak. Wet. Amsterdam, A 59(3), (1956), 338-359.5. N. Jacobson, Lectures in Abstract Algebra, vol. II, New York, 1953.6. C. C. MacDuffee, The Theory of Matrices, Berlin, 1933.7. A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex mani-folds, Ann. of Math., (2) 6 5 (1957), 391-404.

NORTHWESTERN UNIVERSITY

ON THE ZEROS OF THE SOLUTIONS OF

DAVID LONDON

l Introduction. Let/(z) be a meromorphic function with, at most,simple poles in a simply-connected domain D, such that f'(z) Φ 0 forz e D. Let

(1.1) P(z) = i { / ω , *} ,

where

</<*>.*>=(f)'-l(f)"is the Schwarzian derivative of f(z) with respect to z. In connectionwith the function f(z), we consider the differential equation

(1.2) w"(z) + p(z)w(z) = 0 .

The function f(z) may be written in the form

where w^s) and w2(£) are linearly independent solutions of (1.2). Thenontrivial solution

w(z) = i w ^ ) + Bw2(z)

vanishes at zlf z2, * ,zn if, and only if, f(z) takes at these points thevalue — BA'1. Hence, it follows that f(z) takes some value in Dn timesif, and only if, there exists a nontrivial solution of (1.2) having nzeros in D.This connection was pointed out by Nehari in [3].

DEFINITION 1. The equation (1.2) is disconjugate in D if everysolution of (1.2) vanishes in D not more than once.Hence, (1.2) is disconjugate in D if, and only if, f(z) is univalent in Zλ

DEFINITION 2. The equation (1.2) is nonoscillatory in D if every

Received November 9, 1961. This paper represents part of a thesis submitted to theSenate of the Technion-Israel Institute of Technology in partial fulfillment of the require-ments for the degree of Master of Science. The author wishes to thank Professor B. Schwarzfor his guidance in the preparation of this paper. Acknowledgment is also due to ProfessorE. Netanyahu and Professor E. Jabotinsky for their help.

979

980 DAVID LONDON

solution of (1.2) vanishes in D at most a finite number of times.Hence, (1.2) is nonoscillatory in D if, and only if, f(z) is finitely-valentin D.

By imposing restrictions on p(z), disconjugacy and nonoscillationtheorems can be obtained. By the above connection, these theoremsare equivalent to theorems about the distribution of the values of f(z).In this paper these theorems will be formulated as disconjugacy andnonoscillation theorems only, and not as theorems regarding the valuesof f(z).

2. A lemma. The lemma to be proved in this paragraph yields anupper bound for \p{z)\ on \z\ = p, p < 1. This bound is connected withthe area integral I I | p(z) | dxdy.

LEMMA 1. Let p(z) be analytic in \z\ < 1. Then

\p(z)\dxdy

(2.1) \p(z')\

Proof. Let

Then

and therefore

Jo

π(l

P(P

.-\z'\y

P(z) =

eiβ)dθ =

oo

2πa0 , 0 ^ p < 1 ,

Multiplying by pdp and integrating, we obtain

if \p(z)\dxdyJJl2l<1

(2.2) π

which proves (2.1) for zf = 0. The transformation

maps \ξ\ < 1 onto \z\ < 1. Let

(2.3)

ON THE ZEROS OF THE SOLUTIONS OF w"{z) + p(z)w(z) = 0 981

We have

<2.4) [( \p(z)\dxdy = ft \p[z(ζ)]\ f *dξdη = (t |ft

and

<2.5)

Applying (2.2) to the function ^(f), and using (2.4) and (2.5) we obtainthe required result (2.1).

REMARK 1. For the special case when the function p(z) is a squareof a function analytic in \z\ < 1, Lemma 1 can be obtained using theBergman kernel function of \z\ < 1. (see [4], p. 261, ex. 4).

REMARK 2. (2.1) is a sharp inequality. It is easily proved that thesign of equality in (2.1) at a point z' = zo,\zo\ < 1, occurs if (and onlyif) p(z) is of the form

(1 - zzo

3 A sufficient condition for disconjugacy (I) In [3], Nehari provedthe following theorem (Th. 1, [3], sufficient condition): Let p(z) be analyticin \z\ < 1. A sufficient condition for (1.2) to be disconjugate in \z\ < 1is:

This theorem is sharp, as is shown by an example due to E. Hille [2].From Lemma 1 and from Nehari's theorem, we obtain a disconjugacy

theorem, in which the restriction on p{z) is given by a condition on thearea integral.

THEOREM 1. Let p(z) be analytic in \z\ < 1. A sufficient conditionfor (1.2) to be disconjugate in \z\ < 1 is:

<3.2) ^ \p(z)\dxdy £ π .

Proof. From (2.1) and (3.2) it follows that

\\ \p(z)\dxdy 1

\p(z')\<UM<l < ± IzΊ < Ί7 Γ ( 1 - | 2 ' | 2 ) 2 ~ ( l - \ z ' \ y ' | 2 | < . i .

The assumption of Nehari's theorem is satisfied, and therefore (1.2) is

982 DAVID LONDON

disconjugate in \z\ < 1.

REMARK 1. The question of the sharpness of Theorem 1 is still open.Although both, inequality (2.1) and Nehari's theorem, are sharp, it doesnot follow that Theorem 1, which is deduced from them, is sharp too..

REMARK 2. In [7] the following theorem (Th. 4, [7]) is proved: Letp(z) be analytic in \z\ < 1. A sufficient condition for (1.2) to be dis-

con jugate in \z\ < 1 is

(3.3)

The integral on the left-hand side of (3.3) is defined as the limit forp —> 1, of the nondecreasing function

From our Theorem 1, it follows that the constant 4 in (3.3) can beimproved to 2π. Indeed, if

"*\p(eίΘ)\dθ ^ 2ττ,0

then

\2π\p(peiθ)\dθ ^ 2π , p < 1 .Jo

This implies now the validity of (3.2), and therefore, by Theorem 1, (1.2)is discon jugate in \z\ < 1.

The constant 2π is, however, not the best possible. In Theorem &it will be improved to 4ττ.

4. Invariance of the area integral. We shall prove that the areaintegral is invariant under the transformation of (1.2), resulting from alinear mapping of the variable z.

Let

(4.1) ξ = a z + b , ad - be Φ 0 , (? = £ + * ? ) ,cz + d

be a linear transformation analytic in the simply-connected domain D.D is mapped by (4.1) onto D'. By this mapping (1.2) will be transfor-med into an equation of the form

(4.2) W'\ζ) + P{ζ)W\ζ) + Q(ξ)W(ζ) = 0 ,

where

ON THE ZEROS OF THE SOLUTIONS OF w"(z) + p(z)w(z) = 0 983

W(ζ) = w[z(ζ)] .

By the further substitution

(4.3) Wig) = ^ r ,a — ζc

equation (4.2) takes the form:

(4.4) w[\ζ) + P1(ξ)w1(ζ) = 0 .

The solutions w(z) and wx(ζ) vanish in D and D' respectively at pointsz and ξ connected by (4.1). By a simple calculation, or from the prop-erties of the Schwarzian derivative, it follows that

(4.5)

and hence

W e have thus proved that

(4.6) j j^ I p(z) I dxdy = j J P1(ζ) \ dξdη .

The property expressed by (4.6) is the invariance of the area integral.The invariance of the area integral yields the following generaliza-

tion of Theorem 1:THEOREM 1'. Let p(z) be analytic in D, where D is a circle or a

half plane. A sufficient condition for (1.2) to be disconjugate in D is

(3.2)'

Proof. By a suitable linear transformation, D will be mapped ontothe unit circle. From the invariance of the area integral, from Theorem1, and from the fact that the solutions of (1.2) and (4.4) vanish at cor-responding points, the desired result follows.

5 A theorem about zeros on the boundary of a certain domain*Using the invariance of the area integral and a theorem of Grunsky,we obtain a result regarding the zeros of the solutions of (1.2) on theboundary of a domain bounded by two orthogonal circular arcs.

THEOREM 2. Let p(z) be analytic in D, where D is a domain boundedby two orthogonal circular arcs. If

984 DAVID LONDON

(5.1)

then no nontrivial solution of (1.2) vanishes twice on one of the arcsbounding D.

Proof. Let Γ be oneof the two orthogonal arcs bounding D.Assume that there exists a nontrivial solution w(z) of (1.2) and zlf z2e Γ,such that:

wfa) = w(z2) = 0 .

Let A be the domain bounded by the arc z±z2 of Γ, and by the arcpassing through zΊ and z29 orthogonal to Γ and lying within D. LetD2 be the upper half of the unit circle. A suitable linear transformationmaps A onto D2 so that zx and z2 are mapped on ± 1 . As

it follows from the invariance of the area integral that we may assume,without loss of generality, that D is the upper half of the unit circle, andz19 z2 are ± 1 .

We shall make use of the following theorem of Grunsky [1]: Letg(z) be analytic in a convex domain D. Let zlf z2e D be such that

0(Si) = 9(z2) = 0 .

Let A be the triangle with vertices z19 z2 and zr, z' e D. Let A be the areaof Δ. Then

(5.2) 2Ag(z') = {zf - zx) {zr - z2)\\^y\z)dxdy , (z = x + iy) .

From (5.2) we obtain here

2Aw(z') = (zf - l)(z' + ΐ)[[w"(z)dxdy

= - ( * ' - 1) (z9 + l)^p(z)w(z)dxdy ,

and therefore

(5.3) 2A\w(z')\ < \z' - 1| \z' + l | J J jp(^) | \w(z)\dxdy .

Let z* be a point on the boundary of D, at which \w(z)\ takes itsmaximum value in D. There are two possibilities:

I. 2* belongs to the circular part of the boundary of D.II. z* belongs to the diameter of D.

In case I, we get from (5.3):

Noting that

| z * - l | | s * + l | _-,

we obtain:

(5.4)

Inequalities (5.1) and (5.4) are incompatible, proving our theorem in thiscase.

In case II, we use the linear transformation

z + %

Let Df be the lower half of the circle \ξ + i/2| < 1/2. The above trans-formation maps D onto D', so that the circular part of the boundary ofD is mapped onto the diameter of Df, and the diameter of D is mappedonto the circular part of the boundary of Dr. The point z*, which accor-ding to our assumption belongs to the diameter of D, is mapped on thepoint ζ*, which belongs to the circular part of the boundary of Dr. Thepoints z = ± 1 are mapped on the points a = 1/2 — i/2, b = —1/2 — i/2,which are the two endpoints of the diameter of D\ Equation (1.2) istransformed into equation (4.4), for which we have

w^a) — w^b) = 0 .

From (4.3) we get

(5.5) Wl(ζ) - w[z(ξ)](-ξ) ,

and therefore:

••(5.6) M0i<ιrιiiΦ*)i.

Let ξ' — ξ — i/2 be any point on the diameter of ΰ ' . We have:

•(5.7) IΓKIΠ

From (5.5), (5.6) and (5.7) we obtain:

< i n \w(z*)\ < \z*\\w(z*)\ -

As f is any point on the diameter of D', and ζ* is a point on the circularpart of the boundary of Df, we conclude, from the last inequality, that

DAVID LONDON

takes its maximum value in D' on the circular part of the bounda-ry. From the invariance of the area integral it follows that

We have now the same situation as in case I, for which the proof hasalready been completed.

6. A sufficient condition for nonoscillation* Inequality (3.2) wasseen to be a sufficient condition for (1.2) to be disconjugate in \z\ < 1.The following result shows that mere boundedness of the integral ap-pearing in (3.2) is sufficient to assure nonoscillation in \z\ < 1.

THEOREM 3. Let p(z) be analytic in \z\ < 1. A sufficient condition

for (1.2) to be nonoscillatory in \z\ < 1 is:

(6.1) (f \p(z)\dxdy<J JM<1

oo

Proof. Assume that there exists a nontrivial solution w(z) of (1.2)with infinitely many zeros in \z\ < 1. The set of these zeros has anaccumulation point a on \z\ — 1.

From (6.1) follows the existence of a number p, 0 g p < 1, such that

(6.2) ff \p(z)\dxdy^l.

It is obvious that at least one of the two halves of a circle, havingfor diameter the segment connecting two internal points of a given circle,lies inside the given circle.

As a is an accumulation point of the set of zeros, we can choosetwo elements of that set, zx and z2, so that the half circle D, for whichthe segment connecting zx and z2 is a diameter, and which lies in \z\ < 1,will also lie in the circular ring p < \z\ < 1.

Inequality (6.2) implies inequality (5.1) for this half circle D. D isthus a half circle for which (5.1) is satisfied, and on its diameter thereexist two zeros of (1.2). This last fact is a contradiction to Theorem 2,so that no such nontrivial solution w(z) of (1.2) exists.

REMARK 1. In [5] the following theorem (Th. 3, [5]) is proved:Let p(z) be analytic in \z\ < 1. A sufficient condition for (1.2) to be

nonoscillatory in \z\ < 1 is:

(6.3) \2π\p(eiΘ)\dθ< ~ .Jo

(The integral in (6.3) is defined in paragraph 3).

This theorem can be deduced from our Theorem 3. Indeed, (6.3)implies the existence of a bound M such that

\2π\p(ρeίΘ)\dθ <M, O^P<1,Jo

and, therefore, such that

ίί \p(z)\dxdy<^ .JJizKi 2

The assumption of Theorem 3 is satisfied, and therefore (1.2) is non-oscillatory in \z\ < 1.

From the invariance of the area integral follows the validity ofTheorem 3 for every circle and every half plane. In the following theo-rem, Theorem 3 will be extended to more general domains.

THEOREM 4. Let p(z) be analytic in a domain D bounded by ananalytic Jordan curve. A sufficient condition for (1.2) to be nonoscil-latory in D is:

(6.1)' \[ \p(z)\dxdy< oo .

Proof. Let ζ = ψ(z) be a function mapping D onto |f | < 1.

In paragraph 4 we described the transformation of (1.2) by a linearmapping. The transformation of (1.2) by a general mapping ζ = ψ{z)may be performed in a similar way. In the general case we have tochange (4.3) into

<4.3)f W{ξ) = wffie-t^w .

Equation (1.2) is transformed into an equation of the form (4.4), but

(4.5) becomes now

(4.5)'

As the corresponding solutions of (1.2) and (4.4) vanish at cor-responding points, equation (1.2) is nonoscillatory in D if, and only if,equation (4.4) is nonoscillatory in \ξ\ < 1. In order to prove that (4.4)is nonoscillatory in \ξ\ < 1, it is sufficient, by Theorem 3, to show that

(6.1)" if \p1(ξ)\dξdVJJ\ζ\<i

Prom (4.5)' we have

\ζ\<l

988 DAVID LONDON

(6.4) \\{ζJvm\dξdη = j j j p(z) - l{^(z), z} dxdy

, z}\dxdy .

As D is bounded by an analytic Jordan curve, the function ξ = ψ(z) isanalytic in JD, and for zeDψ\z) Φ 0, so that:

(6.5) \\j{ψ(z),z}\dxdy< co .

Inequality (6.1)" now follows from (6.1)', (6.4) and (6.5).

7* A theorem of Pokornyύ In [8] Pokornyi obtained a sufficientcondition for (1.2) to be dίsconjugate in \z\ < 1. This sufficient conditionfollows also from a more general theorem of Nehari (Th. 1, [6]). We givehere an additional proof of Pokornyi's theorem.

THEOREM 5. Let p(z) be analytic in\z\<l. A sufficient condition'for (1.2) to be disconjugate in \z\ < 1 is:

1-1*z\<l..

Proof. Assume that there exists a nontrivial solution w(z) of (1.2),.a n d zu z2, \zλ\, \z2\ < 1, z λ Φ z2, s u c h t h a t

w{zλ) = w(z2) = 0 .

The points zx and z2 determine uniquely a circle passing through themand orthogonal to \z\ = 1. We denote by C the part of this circle within\z\ = 1. We may assume, without loss of generality, that C is in theupper half plane and symmetric with respect to the imaginary axis (see[6]).

Let ip be the point of C on the imaginary axis. The linear trans-formation

1 + ipz

maps the unit circle onto itself, and maps zλ and z2 on ft and p2r

— Kρ1<p2< + 1 . Equation (1.2) is transformed into equation (4.4),,for which

ft) -= Wλ{p2) = 0 .

By (4.5) we have:

dz

dζ

l — i r r i — i

For - 1 < ξ < + 1, we have

I -

and therefore

(7.2)

Let 0 <fJS < 1 be such that -R <*p1 < pl< R. From (7.2) it followsthat:

(7.3)P 2

The strict inequality in (7.3) assures the existence of an e > 0 and ofa neighbourhood D of the segment [pu ft], such that

(7.4) .

From Grunsky's theorem, quoted above, we obtain

(7.5)

The domain of integration Δ is the triangle with vertices at plf ρ2, ζr.A is the area of Δ. Let Db be an ellipse having [plf p2] as its majoraxis, and let the magnitude of its minor axis be 26, b > 0. For a smallenough 6, we have D& c D. Let f6 be a point on the boundary of Db,at which |Pi(ΣΓ)Wi(OI takes its maximum value in D6. From (7.5) itfollows that

2A\w1(ζ1)\ < \ζb - ft I \ζb - ft

Hence,

and therefore

|p,(r.)l >

990 DAVID LONDON

We define the number δb by the equation

(7.7) i Λ a - κ ι = / z a - ir^r + s .It is obvious that δb > 0, and that δb —> 0, for b —> 0. For a small enoughb we have

/rj g\ 2 ^ 2

^ - I f . l ' + i , R2-\ξ»\*

From (7.6), (7.7) and (7.8) it follows that

(7.9) | 2

As ζb 6 D, (7.4) and (7.9) are incompatible, so that no such nontrivialsolution w(z) of (1.2) exists.

8. A sufficient condition for disconjugacy (II). In [4], p. 127, ex.8, the following theorem is mentioned: Let p(z) be analytic in \z\ S 1.Then

S 2JΓ

\p(eiθ)\dθ was defined for functions0

analytic in the open unit circle. It is easily seen that if we use theabove definition for the integral in the right-hand side of (8.1), then(8.1) is also valid for functions analytic in the open unit circle.

From (8.1) and from Theorem 5, we obtain now the following dis-conjugacy theorem:

THEOREM 6. Let p(z) be analytic in \z\ < 1. A sufficient condition

for (1.2) to be disconjugate in \z\ < 1 is:

(8.2) [2π\(p(eίθ)\dθ ^Aπ .Jo

(The integral in (8.2) was defined in paragraph 3).

Proof. By (8.1), for functions analytic in \z\ < 1, and by (8.2), wehave:

| ( ) I 4 • M < 1" v " - 2 π ( l - | z | 2 ) ~ 1 - 1

The validity of (7.1) is thus proved, and therefore, by Theorem 5, (1.2)

is disconjugate in \z\ < 1.

REMARK 1. Theorem 6 improves Theorem 4 in [7]. (see Remark 2in paragraph 3).

REMARK 2. The question whether the constant 4π in (8.2) is thebest possible is left open. That it cannot be improved too much is shownby the example p(z) == ττ2/4. The corresponding equation (1.2) is dis-con jugate in \z\ < 1, and

[2π\p(eiθ)\dθ = — ^ 4 . 9 τ τ .Jo 2

REFERENCES

1. H. Grunsky, Eine Funktionentheoretische Integralformel, Math. Zeitschr., 6 3 (1955),320-323.2. E. Hille, Remarks on a paper by Zeev Nehari, Bull. Amer. Math. Soc, 55 (1949), 552-553.3. Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc,5 5 (1949), 545-551.4. , Con formal Mapping, 1st ed., Me Graw-Hill Book Company, INC., 1952.5. , On the zeros of solutions of second order linear differential equations, Amer.J. Math., 76 (1954), 689-697.6. , Some criteria of univalence, Proc. Amer. Math. Soc, 5 (1954), 700-704.7. , Univalent functions and linear differential equations. Lectures on Functionsof a Complex Variable, Ann. Arbor. (1955), 49-60.8. V. V. Pokornyi, On some sufficient conditions for univalence, Doklady Akademii Nauk

SSSR (N.S.) 79 (1951), 743-746.

TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY, HAIFA, ISRAEL

ON THE RADIAL LIMITS OF BLASCHKE PRODUCTS

G. R. MACLANE AND F. B. RYAN

1. Introduction, As is well known, a Blaschke product f(z) in{\z\ < 1} has radial limits f(eίθ) of modulus one almost everywhere on{\z\ = 1}. The object of the present paper is to give a partial answerto the question: how many times does f(z) assume a given radial limit?We shall prove the following theorem.

THEOREM A. Let E be a given closed set on {\w\ = 1} and let Er

be the complement of E relative to {\w\ = 1}. Then there exists aBlaschke product f(z), all of whose radial limits are of modulus one,and such that the set

has the power of the continuum for eiβ e E and is countable for eίβ e E\

Theorem A is a condensed statement of what we shall actually prove;Theorems 1, 2, and 3 contain somewhat more information on f(z). Themethod of proof is to construct a suitable regularly-branched coveringW" of {\w\ < 1}, corresponding to an automorphic function w — f(z), andthen use the geometry of <W~ to obtain our results.

The question naturally arises as to whether one could prove TheoremA directly. That is: could one produce an/(2) with the desired propertiesby exhibiting its zeros instead of defining/(#) by means of a surfaceThe answer to this question does not seem to be obvious.

2, The surface ^ ~ . Let E be a given nonvoid closed subset of{|w| = 1} and let {αw}Γ be an infinite sequence of points in {\w\ < 1}whose derived set is E. Clearly, we may assume that an Φ 0 and

(1) arg am Φ arg an (m Φ n) .

Let W~ be the simply-connected unbordered covering of {\w\ < 1} whichis regularly-branched over the points {an} with all branch points ofmultiplicity 2. It is well known [2, 3, 6] that such a covering, with anyspecified multiplicity or signature for each an, exists and is unique.Instead of appealing to the general theory of regularly-branched coverings,we shall construct the surface 'W directly, since the details of theconstruction play a role in the proof of Theorem A.

Received September 29, 1961. The research of the first author was supported by theUnited States Air Force through the Air Force Office of Scientific Research of the AirResearch and Development Command, under Contract No. AF 49 (638)-205.

993

994 G. R. MACLANE AND F. B. RYAN

Let Cn be the radial segment arg w = argαw, \an\ ^ \w\ < 1. TheCΛ are disjoint because of (1). We make cuts in {\w\ < 1} along eachCn and so obtain a alit disc W9 copies of which are joined together,according to the following specifications, to form the surface.

0th level. The surface W"ϋ consists of just one slit disc W. Notethat "W^ is simply-connected.

1st level. The surface 5^Ί is obtained by adjoining an infinitesequence of distinct copies of W, namely W{1), W{2), , to "W^ W{n^)is joined to ^ ~ 0 along Cni so as to form a first-order branch-point overαΛ l. The surface ^ " Ί = ^ 0 U \Jn W(n) is simply-connected; for byadjoining the W{n) one at a time we obtain an increasing sequence ofsimply-connected surfaces which exhaust 'Wi We denote by χ(n^) thecurve in "WΊ along which W(n^ and W"Q are identified.

2nd level. Along each free slit on the boundary of W"ι we adjoina copy of W. More precisely, the sheet W(nl9 n2) is adjoined to W{n^)along the cut C%2 in W(n^. The added sheets correspond one-to-onewith all pairs (nl9 n2) of positive integers such that nx Φ n2. Again wesee that the surface <W2 = W"i U U W(nlf n2) is simply-connected. Thecurve over Cn2 along which W(n^ and W(nlftn2) are joined is denotedby χ{nλn2).

kth level. Continuing the construction, the surface W^u consists ofVi and copies of W denoted by W(nl9 n2, , nk), nt Φ ni+l9 which

are joined to W^k-i) W(nl9 * , nk) is adjoined to W(nί9 , nk-ύ alongthe cut Cnjc in W(nl9 , nk^. Denote the curve along which those twosheets are joined by χ(nl9n29 •• ,n Λ ) . Clearly W^u is simply-connected.

We take the surface <W~ to be lim <W*k as k —> co it is clear thatW is simply-connected as Wk ΐ ^ " . With the natural projection maponto {\w\ < 1} it is clear that "W is a regularly-branched, unbordered,covering of {|w| < 1}. All points of ^ over the an are branch-points ofmultiplicity 2, and W" has no other branch-points.

3 The function/(z). Since ^ is a covering of {|w| < 1} it ishyperbolic. Let w = f(z) be the holomorphic function which maps {\z\ < 1}onto 5^\ with /(0) = Oe <W, and /'(0) > 0. Clearly \f(z)\ < 1. Theradial limits of f(z) are all of modulus one, since if this were not thecase a boundary point of {\z\ < 1} would correspond to an interior pointof <%r which is unbordered. Thus/(z) is of class U [5, p. 32]. ApplyingFrostman's theorem [5, p. 33] we see that f(z) is a Blaschke product.

Also, f(z) is an automorphic function with respect to a Fuschiangroup F, since the decktransformations of W" correspond to linear

ON THE RADIAL LIMITS OF BLASCHKE PRODUCTS 995

transformations preserving {|z| < 1}. It is easily shown that if E ={\w\ = 1} then Fis of the first kind: the limit points of F fill {\z\ = 1}.If E Φ \\w\ = 1} then F is of the second kind: the set of limit pointsof F is a perfect nowhere dense subset of {\z\ = 1}.

The sheets W(nl9 n29 , nk) of ^ " correspond to a set of fundamentalregions R(nl9 , nk) of i^. These are the fundamental regions whichplay a role in the proof; since these are defined via the function / itis not clear that they are the same as the fundamental regions obtainedby any of the usual constructions in terms of F. Hence we must derivesome properties of these regions.

4. Properties of the fundamental regions. For convenience wereduce the notations W(nlf •••,%), R(nl9 -- ,nk), and χ(nlf " ,nk) toW, R, and χ respectively. To each curve χ in CW" there corresponds asimple arc X in {\z\ < 1}. It is evident that the fundamental regionsR are bounded by the X's and points of {|z| = 1}. We proceed withan investigation of the X's.

First, each X ends at two distinct points of {\z\ = 1}. The two

linear pieces of χ correspond to two simple arcs X' and X", and f(z)tends to a limit as |«|—>1 on Xf and X". Then by Koebe's lemma[1, p. 213] each of Xf and X" must tend to a definite point of {|s| = 1}.The end points of Xf and X" must be distinct. If not, let D be thatpart of {|21 < 1} bounded by X and a single point b on {\z\ — 1}. Thenthe part of W" corresponding to D will contain an infinite number ofsheets W joined along various χ's, which correspond to X's, all endingat b. Thus f(z) would have infinitely many distinct asymptotic values,namely exp (i arg an), at b; but this would contradict the theorem ofLindelof [4, p. 9] to the effect that a bounded holomorphic function canhave at most one asymptotic value at a given point.

Thus each X is a crosscut of {\z\ < 1}. A second property is thatno two X 's have a common endpoint. To see this, suppose Xλ and X2

are two distinct X ' s with a common endpoint 6 on {\z\ = 1}. Let thecorresponding curves χλ and χ2 in <W end at points aλ and a2, respectively,over {|w| = 1}. If ax Φ a2 then we would again have a contradiction ofLindelof's theorem. Now suppose aλ = a2. We may construct a sequenceof arcs Δn in \\z\ < 1}, each joining a point of Xλ to a point of X2, suchthat diam Δn —* 0. Since by Lindelof's theorem f(z) -^ aλ uniformlybetween Xx and X2 we may also require diam {f(Λn)} < 1/n. But fromthe structure of W" it is clear that there exists a curve χ on ^ 7 , withendpoint φ a19 such that any curve on 5^", joining a point of χx to apoint of χ2, must intersect χ. Since the projection of χ into {\w\ < 1}and the common projection of χx and χ± are a positive distance δ apart,we must have diam {f{Άn)} ^ 3, which is incompatible with diam-{/(//„)} < l/n.

Next, for any ε > 0, the set S = {X\ diam X > ε} is finite. For, anydisc {\z\ < 1 — δ} intersects only a finite number of the X's. Hence ifS were infinite there would exist an infinite sequence {Xn}T of distinctcrosscuts and a nondegenerate arc A on {z\ = 1} such that the radiusjoining z = 0 to an arbitrary point of J crosses every Xn. Now anyradial limit f(eίθ) = eioύ, eiθeA, forces the χΛ, corresponding to Xn andending at eic*n, to satisfy an —> α. But then /(βίθ) = eία> for almost alleiθeA, which contradicts the theorem of F. and M. Riesz. The pointof this paragraph is that if b is a limit point of F, then any neighbor-hood of b contains infinitely many complete fundamental regions R.There are at least some examples of Fuchsian groups possessing aset of fundamental regions (connected) whose diameters are boundedaway from zero.

5 Properties of f(z) on the boundary.

THEOREM 1. Let b be a limit point of F, U a neighborhood of b,and let eioύ e E. Then the set

U n L(a)

has the power of the continuum.

Proof. There exists a cross-cut X, corresponding to the curve χin ^ ~ , which separates {z\ < 1} into two domains, one of which, D, iscontained in U. The corresponding part, £^, of 'W" contains infinitelymany sheets. In {\w\ < 1} we may select among the arcs Cn twosequences, {Cw(0)}Γ, and {Cn(l)}Γ, which satisfy either the following threeconditions(2) the lengths of the Cn(0) and Cn(l) tend to zero,(3) arg Cn(0) i a, and arg Cn(l) [ a,(4) arg Cn+1(0) < arg C»(l) < arg CM;or the same conditions with the arrows in (3) and the inequalities in (4)reversed. Such sequences {Cw(ε)}, ε = 0,1 exist because of ei(* e E, theinitial choice of {an}, and (1).

Now let Γ(ε) = Γ(εly ε2, •••), e< = 0, 1, be an arc in &r with theproperties:(5) Γ(ε) crosses, in order, curves X in ^ over the arcs C^βi), C2(ε2),C3(ε3), •••, and meets no other χ ' s .(6) Γ(ε) tends to a point on the boundary of & over eia.

This construction of Γ(e) is possible by (2), (3), (4), and since allthe curves χ over a < arg w < a + δ, δ = δ(η), are of length < η. Γ(ε)corresponds to an arc Δ(ε) in {\z\ < 1} which tends to a definite pointb(ε)e U Π {\z\ = 1}, since f(z)-*eicύ on Δ(ε). By a well-known theoremof Lindelδf [4, p. 10] then the radial limit of f(z) exists at b(ε) and has

ON THE RADIAL LIMITS OF BLASCHKE PRODUCTS 997

the value of eίa.By associating b(ε) with the dyadic expansion 0. ελε2ε3 , we see

that we have found a set of points b(e) in U Π {\z\ — 1}, associated withthe radial limit ei0C/, having the power of the continuum, provided thatdistinct sequences of ε's correspond to distinct points δ(ε). To show that,let {εj and {ε'} be two distinct sequences and let p be the smallestinteger for which εp ψ εf

p. Then Cp(ep) and Cp(εp) are distinct and thecorresponding crosscuts Xp(ep) and Xp(εp) subtend two disjoint (recallthe structure of W) closed arcs Ap and A'p on U f] {\z\ = 1}. But δ(s)G Ap and δ(ε') e Ap and so δ(ε) φ b(ε').

THEOREM 2. Lei b be a limit point of Fand let Ube a neighborhoodof b. The set

{θ I eiθ e U, f(eίd) does not exist}

has the power of the continuum.

Proof. Select three distinct arcs, C(0), C(l), C(2), from among thearcs Cn. Suppose a curve Γ in "W meets, in succession, curves χ overthe arcs in the sequence

C(eO, C(ε2), C(ε3), . . . (ε, = 0,1, 2; ε, ^ εi+1)

and crosses no other χ's. To those curves χ in W~ which Γ meets therecorresponds a sequence of crosscuts Xl9 X2, X39 , which subtend arcsAlf Λ2, ΛB, ••• on {\z\ = 1} satisfying the condition Aζ+1 c A°n. Also wechoose e1 = 0 and Xx fixed, in 27, so that the image of Γ lies in U.The sequence {εn} then determines a unique point b(e) = [}A°ne U. Theradius to b(ε) intersects all Xn; hence f(z) has no radial limit at 6(e), forC(0), C(l), C(2) are all distinct and εt φ εi+1. Now given the start of thesequence, el9 ε2, , εp, there are two possible choices for εp+1 and thetwo possible arcs Ap+1 are disjoint. Thus distinct sequences {εj yielddistinct points 6(ε). The set of sequences {εj has the power of thecontinuum.

THEOREM 3. Let eiaeE'. Then the set L(a) is countable.

Proof. Let U be a neighborhood of eia containing none of thepoints an. Then ^ c o n t a i n s a countable number of schlicht components^ i , ^<2, ••• over U Π {|w| < 1}. Each ^/ n maps onto y n c {|«| < 1},where Vn is bounded by an arc Anoί {\z\ = 1} and a crosscut of {|s| < 1}The function f(z) is holomorphic on An and there is just one radius,ending on An, associated with the radial limit eioύ. Since Ύ^ con-tains only this countable collection of components over Z7, the result isclear.


We remark that if E is void, then the use of a two-point set {al9 a2}leads to a Blaschke product satisfying Theorem 3. With a three-pointset we can satisfy both Theorem 2 and Theorem 3. Theorem 1 is ofcourse vacuous.

REFERENCES

1. P. Koebe, Abhandlungen zur Theorie der konformen Abbildnng, I. Crelle 145 (1915),177-223.2. , Abhandlungen zur Theorie der konformen Abbίldung, II. Acta Math., 4O(1915-16), 251-290.3. , Uber die Uniformisierung beliebiger analytischer Kurven, I and II, Crelle,138 (1910), 192-253 and 139 (1911), 251-292.4. E. Lindelof, Sur un principe general de Vanalyse et ses applications a la theorie de la,representation conforme, Acta Soc. Sci. Fenn., 46 no. 4 (1915).5. K. Noshiro, Cluster Sets, Ergebnisse der Mathematik and ihrer Grenzgebiete, n. F. 28,Springer, Berlin 1960.6. K. Reidemeister, Einfύhrung in die kombinatorische Topologie, Vieweg, Braunschweig1932.

RICE UNIVERSITY

EVALUATION OF AN ^-FUNCTION WHEN THREE OFITS UPPER PARAMETERS DIFFER BY

INTEGRAL VALUES

T. M. MACROBERT.

1. Introduction. If p ^ q + 1, [1, p. 353]

. Γ(ar + »)ΠT(α, -ar-n)(1) E(p; ar: q; ps: z) =

n\ s -ar-n)

where, if j) = g + 1, | 2 | < 1. The dash in the product sign indicatesthat the factor for which t = r is omitted.

Now, if two or more of the α's are equal or differ by integralvalues, some of the series on the right cease to exist. For instance,if ax — a + ϊ, a2 = a, where I is zero or a positive integer, it has beenshown [2, p. 30] that the first two series can be replaced by the ex-pression

Γ(a + I + -a-l-n)

(2)

n!(Z + n)lf[Γ(ps -a-l-n)

+ n)(i - n - l)\ΐlΓ{at - a - n)^ (

s -a-n)

where

= ψ(ϊ + w) + ΨW - ψ(a + 1 + n -1) — logz

ί ( a t - a - l - n - 1 ) -Σψ(ρs - a - l - n - 1 ) .

Here

(3)

so that

(4)

az

It will now be shown that, in the case in which

Received September 28, 1961.

999

1000 T. M. MACROBERT

aλ — α, a2 = a + I, a5 = a + I + m ,

where Z and m are zero or positive integers, the first three series canbe replaced by the expression

Γ(a + 1 + m + :, — a — I — m — n)(-z)n

(5)

(Σw=0

m + ri)\γ[Γ(ps — a — I — m — ri)s = l

„_/(« + I + w)(m - n - l)!ΠΓ(α ( - a — I - n)zn

Σ — 6

n!(Z + n)!ΠΓ(|θβ - a - I - ri)

-m-n-l)lflΓ(at-a-n)- n -

x

where

n)'^ log £ + ^(*x + Z + m + τi — 1) — ψ(ϊ + ΎΪI + ?2/) — ^(?9V

— ψ(ri) — Σ ^ ί ^ ί — a — I — m — n — 1)ί=4

+ ΈiΨiPs — a — I — m — n — 1)s=l /

+ Z(α + l + m + n — 1) — χ{l + m + ri) — χ(m + ri) — χ(ri)

+ ΣiXiat - a - l - m - n - l ) - Σz(i° s -a-l-m-n-1),ί=4 β=l

and

©TO = log z + ^(<x 4

ί=4

Here

(6)

where | ampz\ < π.

n — 1) —q

. "=1

ri) — ψ(m — n — 1) —

- a - I - n - 1) .

Σ_i_,

2. Proof of the formula. If aλ = a, a2 = a + I, a3 = a + ϊ + m + ε,where i and m are zero or positive integers and e is small, it followsfrom (2) and (1) that the first 3 terms of (1) are equal to A + B + C,where

EVALUATION OF AN ^-FUNCTION WHEN THREE OF ITS UPPER 1001

v«, Γ(a + I + n)Γ(m - n + ε)Π^(^ί — a — l — ri)

A = (-l)^ α + z Σ - — ^»«n ,nl(l + n)!ΠΓ(/o. - a - I - n)

"where

— l o g z + Σ Ψ O ^ ί "- a — I — n — 1 ) " " ΈtΨiPs — oc — I — n — 1 ) ,ί=4 s=l

- m — w + sJΠΠ^ί — a — n)

lilΓ(ps -a-n)s=l

Γ(α + I + m + n + ε)Γ(— I — m — n — ε)Γ(— m — n — ε)

wIΠ^ίΛ — a — I — m — n — ε)s=l

x Π ^ ( ^ ί — α — Z — m — — ε ) x ( — z ) n .ί=4

Then A = D + E, where

or. 7-T/ _ . i 7 m + n)Γ(-n + ε)J{Γ{at -a-l-m-n)at ( m + n)\{l + m + n)\JlΓ{ρs — a - l — m — n )

L n

"where

+ n) + ψ(m + n) + ψ(—n — 1 + ε)

— o/r (α + l + m + n —1) — log « + Σ ^ ί ^ ί — OL — I — m — n —

s — a — I — m — n — 1).

β. In these formulae ί takes the values from 4 to p.Here, since [2, p. 31]

<7) ψ(~z - 1) = f (z) + πcot πz ,

<8) τ/r(—n — 1 + ε) = τ/r(w — ε) — π cot πε .

Hence

C + £7 - _JϊL_(_i)*3-+*+»sm2 πε

1002 T. M. MACROBERT

x

,Γ(α

n\Γ(m + n + 1 + ε)YlΓ{at -a-I - m - n - e)(-z)n

Γ(l + m + n + 1 + ε) x ΐ[Γ(ρs - a - l - m - n - έ )

sin πε , Γ(a + I + m + n)ΐ[Γ(at — a — I — m — n){—z)n

π

x

=oΓ(n + 1 — ε)(m + n)l(l + m + n)lΐ[Γ(ρs — a — l—m — n)

m + n) + α/r(m + ) + ( ^ — ε) — ψ(a

—log « + ΣiΨ(at — oc — I — m — n — V)

_ c o s π£

)— a — I — m — n —

- a: - Z - m - ^)(-g) n

ε)(m + n)l(l + m — m —

The limit of this function when ε —> 0 is obtained by replacing*π2/sin2 πε by i, finding the second derivative with regard to ε of theexpression in the large bracket, and then making ε —• 0. It is

mt -a-l-m- n)(-z)n

x

*=on!(m + w)!(£ + m + n)!Π^(i°. — # — £ — m — %)

log £ + τ/r(α + £ + m + w — 1) — ψ(m + n) — ψ(l + m + n)

— Yiψfat — oc — I — m — n — 1) + ΣΨiPs — oc — I —m — n — 1)

+ χ(a + I + m + n - 1) - χ(m + w) - χ(l + m + n)

+ Σ z ( α ί — <x — i — m — ~ 1 ) — Σz(ι°s ~~ α ~ " I — m — n — 1)

τ^(ί + W + 'λZ') + ijr^Wf + %) + ijf(7l)

o . N - f ( α + i + m + n- 1)(7 w - logz + Σtfe -α-i-m-w-1)

~~ΈιΨ(Ps — oc — I — m — n — X))- 2χ(n) + τr2 - {f (w)}2 + χ(^)

From this, with B and D, formula (5) is obtained.

REFERENCES

1. T. M. MacRobert, Functions of a complex variable, London, (1954).2. T. M. MacRobert, Evaluation of an E-function when two of the upper parameter®differ by an integer, Proc. Glasgow Math. Assoc, 5 (1961).

T H E UNIVERSITY,

GLASGOW

THE SPECTRA OF MINIMAL SELF-ADJOINT EXTENSIONSOF A SYMMETRIC OPERATOR

ROBERT MCKELVEY

1. Introduction. Let T be a closed symmetric operator with do-main Dτ dense in a Hubert space 3ίf. A (generalized) spectral resolu-tion of T is a family of bounded self-adjoint operators Eμ. defined for— oo < μ < oo and such that:

(a) Eμ. is nondecreasing, continuous from the right, and i?-*, =0, £L - 1.

(b) For ueDτ and v e

{Tu, v) = [°° μd{E^u, v), \\ Tu ||2 = Γ μ2d(Eμu9 u) .

When in particular T is self-adjoint, it possesses only one generalizedspectral resolution, namely the orthogonal spectral resolution where Eμ.is for each μ an orthogonal projection. For an account of the theoryof generalized resolutions see [1], Appendix I.

M. A. Naimark has shown that for each generalized resolution Eμ.there is at least one self-ad joint extension Γ+ of Γ in a Hubert spaceJg^f D Sff with the following property: If E£ is the orthogonal reso-lution of T+ and P is the projection onto the subspace έ%f of ^g^+,then Eμ. = PE£. We shall usually require that T+ be a minimal self-adjoint extension of T, i.e. that J%*+ be the closed linear hull of theset of vectors E£3l?9 (—00 < μ < 00); (see § 3). The minimal extensionT+ corresponding to a given Eμ. is determined by Eμ. uniquely, up tounitary equivalence ([8], § 4). We shall denote it by T+ = φ(Eμ).

In this paper we investigate certain questions regarding the spectrumΣ of T+ = ψ(Eμ). In view of the above mentioned unitary equivalence,the point set Σ depends only upon Eμ.\ it may in fact be characterizeddirectly as the set of points of increase of Eμ. (see § 3). Parts of thespectrum—e.g. eigenvalues and essential spectrum—may likewise becharacterized directly in terms of Eμ. It will be convenient to referto the spectrum of T+ as the spectrum of Eμ.

We are interested in comparing the spectra of various resolutionsof a given T. In order to describe the situation precisely, one refersto A. V. Straus' extension theory of symmetric operators [10]. For anycomplex λ, let AT{X) denote the range of T — λ. By definition, thedefect subspace M{X) is the orthogonal complement in 3$f of JT(X).

Received July 12, 1961. This work was carried out at the University of California atLos Angeles during Spring Semester 1961 and was supported by the U.S. Office of NavalResearch.

1003

1004 ROBERT MCKELVEY

Straus has associated with each generalized resolution Eμ of Γ a familjrof contraction operators Fκ, mapping M{i) into M(—ϊ), and such thatFκ is analytic on J^X > 0 with \\FK\\ ^ 1 there. Conversely each suchfamily of contractions is associated with some Eμ. A constant unitaryF corresponds by this association to an orthogonal resolution Eμ, andfor these Straus' extension theory reduces to that of J. von Neumann.(For a complete description, see § 2)

We characterize the spectral resolutions Eμ of T by the behaviornear the real axis of the corresponding Fλ. Specifically we single outtwo extreme cases, where Fλ satisfies, respectively, conditions a and β'or condition γ as defined in § 4. These are local conditions, defined foran open real interval Δ. When Eμ is an orthogonal resolution, condi-tions a and β hold on the entire real axis.

In §§4-6 we consider a symmetric operator T with equal finitedefect numbers (n, n). In § 4 we extend to generalized resolutions ofT, satisfying conditions a and β on an interval A, the theorem of H.Weyl [13] on the invariance of essential spectrum. In §6 we obtain a.parallel theorem on the invariance of absolutely continuous spectrum,proved for T a singular second order ordinary differential operator. Thisextends a theorem of N. Aronszajn [2]. (The theorems of Weyl andAronszajn both concern self-adjoint extensions of T in Sίf, hence or-thogonal resolutions.)

When Fλ is such that a and β fail everywhere on an interval Δan altogether different pattern emerges, for in this case Δ lies entirelywithin the spectrum of Eμ. In § 5 we adopt the more stringent as-sumption that condition γ holds on Δ. In particular, suppose Fλ is afamily of strict contractions, i.e. satisfies condition 7 on the entire realaxis. Suppose that T — μ has a bounded inverse for each real μ. ThenT+ = ψ(Eμ) is unitarily equivalent to the n-ϊo\ά direct sum of iD withitself, where D is the differentiation operator in L2 (—°o, oo). Thisgeneralizes a theorem proved by Coddington and Gilbert ([4], Theorem 14)for T a regular ordinary differential operator of order n. As is indi-cated in § 6, the situation is more complicated when T is a singulardifferential operator.

The study of the spectrum of Eμ requires an analysis of the be-havior of the resolvent Rκ of Eμ near the real axis. The generalizedresolvent Rκ of a spectral resolution Eμ is defined for ^ \ Φ 0 by

(1.1) Rλ =

Thus Rλ is a bounded operator with domain Sίf, analytic on each halfplane J?X > 0, ^X < 0. Inversely, Eμ is determined by Rκ throughthe formula

THE SPECTRA OF MINIMAL SELF-ADJOINT EXTENSIONS 1005

(1.2) {E{Δ)u, u) = lim — ( ^(Λμ + i βw, u)dμ

where Δ is an interval (μlf μ2], μx and μ2 are continuity points of Eμ,and E(Δ) = E^ — EH. When T is self-adjoint, the (generalized) re-solvent jRλ of its orthogonal resolution Eμ. coincides with the resolventof T, i.e. Rλ = (T - λ)"1 for J?X Φ 0.

Let T+ = α/r(^) The resolvents i2λ

+ and i?λ of J5+ and 2£μ arerelated, when ^X Φ 0, by (see [1])

(1.3) R, =

A. V. Straus [10] has given another characterization of Rk, when0, as the resolvent of a certain quasi-self-adjoint extension Tλ inof T. (For precise definition, see §2). Thus

(1.4) R, - (Γλ - λ)" 1 .

In §2 we investigate limit values, as λ tends to the real axis, ofJKλ. It is found that in general the interpretation (1.3) fails for limitvalues while (1.4) retains its meaning. The interpretation (1.3) remainsvalid on a real interval Δ precisely when jBλ can be continued analyti-cally through Δ, and this is possible precisely when Δ lies in the com-plement of the spectrum of Eμ (theorem 3.1).

It is a pleasure to express here my indebtedness to E. A. Coddington,who first drew my attention to generalized resolutions and in particularsuggested that the theorem of Coddington and Gilbert, referred toabove, might be valid in a broader setting. During the course of thework I have had access to his library and frequent benefit of hiscounsel.

2, Limit values of the resolvent We shall designate an arbitraryone of the half planes J^X > 0, J^X < 0 by π+ and the other by π~.Choose any λ o e π + and any contraction operator F (i.e. | | . F | | ^ 1) withdomain M(X0) and values in Af(X0). The operator Γ, defined by

(2.1) ΓcfcΓ*,D$ = {u:u = u0 + φ — Fφ. u0 e Dτ,φe M(x0)}

has been called by A. V. Straus a quasi-self-adjoint extension of T.The class C+ of operators f obtained by holding λ0 fixed and varyingF is, in fact, independent of the choice of λ0 e π+. (See Straus [10],Lemma 9 and the discussion preceding it). A second, and in generaldifferent, class C~ of quasi-self-adjoint extensions of T is obtained bytaking λ0 e π~.

Let Rλ be the resolvent of Eμ of T. Straus has proved that, to


eachλeτr + corresponds a quasi-self-ad joint extension TλeC+ such that

(2.2) (Γ λ - λ)-1 = B λ , λ e π+ .

For a fixed choice of λ0 e π+, the corresponding contraction Fκ = FK(\Q)is analytic in λ on π+. Conversely, any analytic contraction Fκ carryingM(X0) into M(λ0) gives rise, through (2.1) and (2.2) to a resolvent i?λ

of T. The relation R-λ = R? (which follows from (1.1) or (1.3)) has asits correspondent the relation

(2.3) F λ(λ0) - [Fλ(λ0)]*

defining a contraction taking Af(λ0) into ikf(λ0).The following theorem shows that these statements remain valid in

a limiting sense on the real axis.

THEOREM 2.1. Let λx, λ2, in π+ tend to λ on the real axis.(A) Suppose that for a certain λ0 6 π+ the sequence of contractions

F\k(\) converges in norm as k —• co. Then the same is true for everyλj G τr+. The limit, also a contraction taking M(λ0) into M(X0), will bedenoted by FA

λ+ = F\+(X0). It defines a quasi-self-adjoint extension inC+ of T, and the extension Tχ+ so obtained does not depend upon theparticular λ0 e π+ figuring in its construction.

(B) Necessary and sufficient for the convergence in norm of RK]6

to a limit, denoted by Rχ+, is:(i) Convergence in norm of Fλ]c, and(ii) Existence of (TA

λ+ — λ)-1 as a bounded operator with domain £ίf.In this case,

(2.4) RU = (Tu - λ)-1 .

(C) In any subset of [π+ plus the real axis] in which both i?λ andFκ are defined (by extension), the single-valuedness and continuity ofeither implies that of the other.

(D) When, as above, Rλjc and Fλjc(XQ) tend to limits in norm, thenthe same is true of R^k and Fj_Jx0), and

(2.5) Rx- = [RuV , *V(λ 0) - [Fu(λ 0)]* .

Proof. (A) Let ΐ^(λ0) denote the Cayley transform of a quasi-self-ad joint extension f of T. Thus W= U@F, where

C7(λ0) = (Γ - λo)(Γ - λo)"1

is the Cayley transform of T. One easily shows ([10], equation (5.22)),that for λ0 and λj in 7Γ+,

(2.6) W(K) = [(λί - λo) - (λj - λo)ΐ^(λo)][(λί - λo) - (λί - λo)ΐ7(λo)]-1

where the inverse shown is a bounded operator with domain Sίf. Sincethis equation holds between Wλjc(X0) = U(X0) φ Fλjύ(X0) and WλJXΌ), there-fore by continuity WΛ

λ+(K) = lim Wλjc(X'o) exists. Furthermore WΛ

λ+(X0)and W\+(X'Q) are related by (2.6) and hence are Cayley transforms rela-tive to λ0 and λί, of the same f = Γ λ + . Since Wλjc(K) = U(X'Q) © Fλjc(X'o)therefore Fu(K) = Km Fλ]c(X'o) exists, and TFλ+(λί) = W ) 0 F λ + W ) .Thus FA

λ+(X0) and F\+(X'O) define the same extension of Γ.(BX Here we establish the necessity of the condition. Let λ0 e π+.

It follows from (2.2) that, for λ e ττ+,

2\ - λ0 - (Γ λ - λ) + (λ - λ0) = [1 + (λ - λo)# λ](T λ - λ)

and therefore that

(Γ λ - λo)-1 = Rλ[l + (λ - λo)^]" 1 , λ e π+ .

Here [1 + (λ — λo)jBλ]-1 is bounded with domain §ίf. ([10], equation

(5.30), footnote.). By assumption, λ^-^λ on the real axis, and Rλjc-^Ri+in norm. By choosing a special λ0 for which | λ — λo | || RA

λ+ || < 1, weguarantee that [1 + (λ — λo)i2^+]"1 exists, is bounded, and has domainJ%f. Consequently the operator

Gλ = Rλ[l + (λ - λo)^]- 1

is well defined for λ = λ + as well as X e π+, and Gλjc—>G\+ in norm.The Cayley transform Wλ(X0) of 2\ for λ e 7Γ+, is given by

Wλ = (T λ - λo)(Tλ - λo)-1 = 1 + (λ0 - λo)(Γλ - λo)-1 .

Hence

(2.7) Wλ = 1 + (λ0 - λo)Gλ , for λ e π+ .

We define the transformation Wχ+ also by this formula, and show thatWχ+ is a quasi-unitary extension, with || Wλ+ || ^ 1 of the Cayley trans-

form £7(λ0) of T. In fact, the statements

II Wλ || ^ 1 WJ= U(XQ)f for / e Jτ(\)

are valid for λ e 7Γ+ and, since by (2.7) TFλjfc —• PΓA

λ+» a r e valid for λ =

λ + as well. But by [10], Lemma 8, these statements imply that WA

λ+

is a quasi-unitary extension of U(X0).Consequently Wχ+ is the Cayley transform of a quasi-self-adjoint

extension (of class C+) of T. From the relation

X for

it follows, since Wλjc —• W\+, that: JP^ + = limi^7^ exists and

Wu=


Thus Wχ+ is the Cayley transform of the extension which we havedenoted (in A) by Tχ+.

From the relation between any quasi-self-adjoint extension and its-Cayley transform we have

Wu = (T% - \)(Tu - λo)-1 = 1 + (λ, -

Comparing this relation with (2.7), we conclude that

/ rn. -\ \—1 /"^ * Z>A M i /C~ Λ \ D A 1—1

l^ λ+ ~ λo; = {jrχ+ = iίλ+L-L "T (Λ ~ λojiίi+j

From this it immediately follows that R^1 exists and that

TV - λ0 = [l + (λ - \0)Rχ+]Rχ+- = R^1 + (λ - λ 0 ) .

Hence

or

JA -_ CJ1* λ) - 1

This shows the necessity of conditions (i) and (ii) for the special choiceof λ0 made in the course of the argument. But from part (A), alreadyproved, it follows that the conditions hold as well for any other λ0 G π+.

(B)2 In order to prove the sufficiency of the conditions (i) and (ii),we make use of the inverse relation between Tλ and its Cayley trans-form, namely

T λ = (λ0ΫFλ - λo)(ΐfλ - I )" 1 for λ G π+ or λ = λ + .

(For notation, see the proof of part A). Hence

Tλ - λ = [(λ0 - X)Wλ + (λ - λo)](T^λ - I)- 1 .

and, since (2\ — λ)"1 exists (condition ii), therefore

(2.8) (Γ λ - λ ) - = (Wλ - l)[(λ - λΌ) - (λ -

for λ G π+ or λ = λ + .

Furthermore, since the inverse appearing on the left side of this equa-tion is bounded with domain 3$f, the same is true for the inverseappearing on the right side. This fact, together with Wλjc —> WA

λ+ innorm, shows that

which proves the proposition.(C) This is a direct consequence of the reciprocal relations (2.7)

and (2.8), namely


Rλ = (W, - l)[(λ0 - λ) - (λ - λo)^]- 1

Wλ = 1 + (λ - λo)Λλ[l + (λ - λo)^]"1 .

These are valid in 7Γ+ and under the assumptions of (C), are valid, inthe limiting sense, on the entire set considered. Since the inversesdisplayed are bounded operators with domain £ίf, the assertion regard-ing continuity is evident.

REMARK 2.1. When Rχ+ exists (as a limit in norm) it is, by Theo-rem 2.1, an extension of (T — λ)~\ This implies that λ is a point ofregular type of T, i.e. that (T — λ)"1 exists and is bounded.In particular (see [1], Chap. 7), the defect numbers of T are equal.

REMARK 2.2. Necessary and sufficient for the continuity of Rκ

across an open interval Δ of the real axis is:(i) Continuity of Rκ down to Δ in π+ and(ii) Self-adjointness of Rλ+ on Δ, i.e. i?λ+ = i?λ_.

In the presence of (i), condition (ii) is equivalent to(ii)' Unitariness of Fκ+ on Δ, i.e. (Fλ+)-λ = F λ_.

Under these conditions iϋλ is in fact analytic across Δ.(One has only to consider (i2λ/,/), which is analytic in π+ and π~~ andcontinuous across Δ)

3 Resolvent set and spectrum* By the resolvent set of a spectralresolution will be meant the points of 7Γ+ U π~ plus any real point λ0

contained in an open real interval Δ across which Rλ may be continuedanalytically. The resolvent Rλ at λ = λ0 is the common value of thelimits Rλ0+ and Rλ0- there.

In this paragraph we characterize the resolvent set, showing thatit is the complementary point set of the spectrum of EM described inthe introduction.

According to M. A. Naimark, the spectral family Eλ in £ίf may beregarded as the projection on £έf of an orthogonal family E£ in anenclosing space ^ + 3 £ίf. Thus Eλ = PE£, where P is the orthogo-nal projection onto 3ίf\ P ^ + = £$f. The family E£ is the spectralresolution of a self-ad joint operator T+ in Jg^+. In the following weshall assume that T+ is a minimal self-ad joint extension of T, thuswe assume that the set of vectors

{E+(Δ)h: Δ is any interval, h e

is fundamental in ^g^+. In other words, ^f^ is the closed linear hullof this set. (See Naimark [8], §4).

LEMMA 3.1. Let Δ he a (possibly degenerate) interval of the real

axis. Then(A). The set of vectors

Z{Δ) = {E+(Δr)h: Δ' c Δ, h e

is fundamental in E(B) ί?+(J) = 0 if and only if JE7(J) = 0.

Proof. (A) Given / e E+(Δ)^P+. For any ε > 0 there exists g =Σ*-i E+(Ak)gk, for certain intervals Λ and certain #& e £(?, such that11/ - g \\< e. We can write E+(Δk)gk = E+(Δk n )<7* + #+(Λ - Λ)Λ,and thus g = #(1) + #(2) with <;(1) = Σ £ i E+(ΔfM] and </(2) - Σ^iE+(Δr/)gf,where JJ c ^ and Δ'j Π J = 0. Thus #(1) e E+{Δ)£έf+ while #(2)

1 J B + ( J ) T + ,

and | | / - ^ | | 2 - | | / - ^ ( 1 ) H 2 + l k ( Ί ! 2 . It follows that 1 1 / - <?(1)||< s,proving the proposition.

(B) (i) Suppose E(Δ) > 0. The there exists h e Sίf such that0 < (E(Δ)h, h) = (PE+(Δ)h, h) = (E+(Δ)h, h) = \\ E+(Δ)h\|2. Thus E+(Δ) > 0.

(ii) Suppose E(Δ) = 0. Then for Δ' c Δ, E(Δ') - 0 also. Hence forh e Sίf, 0 = (JS?(4f)A, λ) = (E+(Λ')h, h), i.e. JS?+(J')λ = 0. By part (A)this implies that E+(Δ) = 0.

THEOREM 3.1. A real point λ of the resolvent set of the spectralfamily Eλ of T may be characterized in these equivalent ways:

(A) Rλ may be continued analytically across some open real inter-val Δ containing λ.

(B) E(Δ) = 0, for some real interval Δ containing λ.(C) λ is in the resolvent set of a minimal self-adjoint extension

T+ = f(Eλ) of T.In this case, Rx = PRA

λ

+, where iϋλ

+ is the resolvent of T+.

Proof. (A—>B) This is a consequence of the formula (1.2).(B —> C) By the lemma, E+(Δ) = 0. Since El is an orthogonal

resolution of the identity, this implies that the points of Δ are in theresolvent set of T+.

(C—*A) If Δ is in the resolvent set of T+ then RΛ

λ

+ exists forλ G Δ, and PRχ is well defined for points in Δ as well as for nonrealpoints. Since R£ is analytic across Δ, the same is true of PR£. Butfor nonreal λ, Rλ = PR£. Hence Rλ can be continued analyticallythrough Δ, and will then equal PR£ there.

EEMARK 3.1. The representation Rλ = PR£ throughout the resolventset allows the establishment of a number of formulas already knownfor nonreal points:

μ — λ

(ii) For / 6 ΔT{X), (Λμ - Rλ)f = (μ - X)R,Rλf(iii) Δτ{μ) = [1 + (λ - μ)Rk]Δτ(X).We next obtain a result concerning the point spectrum of a minimal

self-ad joint extension T+ of T. In the following theorem, dim g7 de-notes the dimension (^oo) of the manifold i?(λ) of solutions of T*u —Xu. Also E[X] = Eλ+ - # λ _ .

THEOREM 3.2. Let M+(x) be the characteristic manifold incorresponding to an eigenvalue X of T+, a minimal self-adjoint ex-tension of T. Then dim M+(X) = dim E[X\£ί? g dim gf (λ).

Proof, (i) E^κ\3ίf c S?(λ); proving the inequality in the theorem.To verify this let fce^T and choose / e Dτ. Then I7/ = T+f, and(#[λ]Λ, Γ/) = (E+[X]h, Tf) = (T+E+[X]h,f) = (XE+[X]h,f) = X(E[X]h,f).Thus J£[λ]fc e Dτ* and Γ * ^ ^ ] ^ = XE[X]h.

(ii) By Lemma 3.1, E+[X]£^ is dense on Λί+(λ). Thus dim M+(X) =dim E+[X]β£*. The theorem will be proved by showing dim E+]X]βέf =dim E\x\£έf.

Suppose Λ, ,/ w are vectors in T such that JS'+lλ]/;, ., £ f+[λ]/m

are linearly independent. Then E[X]fly , E[X]fm are also linearlyindependent. For otherwise there would be constants cl9 •• ,cm, not allzero, such that

ckE+[X]fk = Σ c^[λ]Λ = 0 .

This would then imply that / = Σ c/^+lΛ]Λ was a characteristic vectorof Γ + such that / e < ^ + θ ^ B ^ t that cannot be, since no reducingmanifold of a minimal extension can lie in g^+ Q έ%f (see Naimark [8],§4.)

On the other hand E+[X]flf , E+[X]fm are obviously independentwhen their projections E[X]flf , E[X]fm are. Thus dim Edim E[X]β^, proving the theorem.

REMARK 3.2. Because of the unitary equivalence of all minimalself-ad joint extensions T+ associated with a given spectral resolution Eμ.of T, it is natural to associate with Eμ the various aspects of thespectrum of T+. Thus by the spectrum, point spectrum, essentialspectrum, etc. of E^ will be meant the corresponding point sets in thespectrum of Γ+. An eigenvalue of E^ will mean an eigenvalue of T+,with its multiplicity the dimension of the corresponding manifold in Jg^+.

From the theorems of this paragraph it follows that certain aspectsof spectrum may be simply characterized directly in terms of Eμ. Wemention especially:

( i ) Spectrum: The points of increase of E^

(ii) Eigenvalues: Points of jump of Eμ. The multiplicity of aneigenvalue λ is dim E[X\β^.

(iii) Point Spectrum: Closure of the set of eigenvalues.(iv) Essential Spectrum: Cluster points of the spectrum, plus

eigenvalues of infinite multiplicity.

4«. Essential spectrum* Let Eμ be a generalized resolution of theidentity associated with a symmetric operator T. From Remark 2.2, anecessary condition for an open real interval Δ to belong to the re-solvent set of Eμ. is that the associated family of contractions Fk fromM(X0) to M(X0) have the properties:

(a) Fλ is continuous from π+ down to Δ, and(β) Fk+ is unitary on Δ.These properties obviously cannot hold for any Δ unless the defect

spaces ikf(λ0) and M(X0) have the same dimension. Hence, when T hasunequal defect numbers, the spectrum of any resolution i?μ consists ofthe entire real axis.

On the other hand, when T has equal defect numbers the properties(a) and (β) may well hold; in particular, when Fκ is a constant unitaryoperator, thus when Eμ is an orthogonal resolution, the properties arevalid for every interval Δ.

In the remainder of the paper we shall consider a symmetric oper-ator A with equal finite defect numbers. We recall that the essentialspectrum Σe is the same point set for all orthogonal resolutions of A,that is, for all self-adjoint extensions in Sίf of A. This is the classicaltheorem of H. Weyl, ([13] p. 251), proved originally for ordinary differ-ential operators, and later extended to abstract operatars by E. Heinz[6]. The principal theorem of this paragraph extends WeyΓs result togeneralized resolutions which satisfy (a) and (β).

THEOREM 4.1. Let the symmetric operator A have defect numbers{n, n) with n < oo, and let Σe denote the points of the essential spectrum* ofany (hence every) orthogonal resolution of A. If Eμ, be an arbitrarilychosen (generalized) resolution of A with essential spectrum Σ'e, then:

( i ) Σ'.ΊΪΣ..

(ii) When (a) and (β) hold on Δ for the family of contractionsassociated with Eμy then Σ'e and Σe coincide on Δ.

(iii) If (oc) and (β) fail on every subinterval of Δ, then Δ c Σ[.We remark that the hypothesis of (iii) holds in particular under the

condition:(γ) Fλ is continuous from π+ down to the open real interval Δ

and \\Fλ+\\ < 1 on Δ.

The proof will be based upon two lemmas of independent interest.

Por any complex λ, let i?(λ) denote the eigenspace of solutions ofT*u = Xu. Thus gf (λ) = Λf(λ).

LEMMA 4.1. Let T be a quasi-self-adjoint extension of T definedby F: M(λ0) —> M(X0). For f,geDτ* introduce the form </, g} =(ϊ7*/, βf) - (/, Γ*flf). Then the domains of t and f* have the following•characterization:

Di = {u: ue DΓ and <u, φ - F*φ> = 0 for all φ e %f(XQ)}

Dfa — {u: u G DΓ* cmc? <w, ψ — Fψy = 0 / o r αii ψ e §?(λ0)} .

Proof. The proof of Theorem 1 in Coddington [3] is directly adap-table.

LEMMA 4.2. Consider a symmetric T with equal finite defectnumbers (n, n), and suppose that X is a real point of regular type ofT, i.e. that T—X has a bounded inverse. For any quasi-s.a. extensiont, if (f — λ)"1 exists, it is a bounded operator with domain

Proof, (f - X)-1 is defined on 4f>(λ) = JΓ(λ) 0 [4# (λ) θ 4r(λ<)] Itis bounded on the first since (T — λ)"1 is bounded at a point of regulartype, and bounded on the second since the enclosing subspace M(X) hasdimension n. Hence (f— λ)"1 is bounded on the sum of these orthogo-nal manifolds.

It remains to show that z^(λ) = £$f. Since Δτ{\) is closed, theproblem reduces to showing that z/ (λ) Q JT(X) is ^-dimensional. By(2.1), which gives the domain of t, and by the existence of (f — λ)"1,it follows that ^ί(λ) contains n vectors which are linearly independentmod JΓ(λ). Their projections onto J#(λ) © r(λ) are therefore linearlyindependent. Q. E. D.

Proof of Theorem 4.1. The statement that in general Σ\ Z) Σe

follows from a result of Hartman, ([5], §3, proof of proposition (iii)):He has shown that, when X e Σe (and n is finite), there exists a sequencefn e DA such that | | / Λ | | = l,/w-*0 weakly (in Jg^) and (A - λ)/w —0strongly. Consequently for any extension A+ in β^+

f fn e DA+, fn —> 0weakly (in Jg^+), and (A+ — X)fn -> 0 strongly. Thus by WeyΓs cri-terion ([9], § 133), λ is in the essential spectrum of A+, and (by Remark3.2) in the essential spectrum of the corresponding Eλ.

Next we show that, under the conditions (ii) on Fλ9 when X $ Σe

it cannot belong to Σ'β. Since X $ Σe, therefore the eigenspace of A atλ is finite dimensional at most. We can depress 3ίf and every Jg^+

to the orthogonal complement of this manifold without changing anyessential spectrum. Hence it may be assumed from the beginning that

λ is not an eigenvalue of A. Hence, by [5], §3, property (ii), thereexists a self-adjoint extension A in §ϊf of A for which λ is not aneigenvalue. Since X $ Σe, it cannot be a cluster point of the spectrumof A; consequently λ is in the resolvent set of A. Let Δ about λ be

o o

an open real interval in which Rx — {A — λ)"1 is analytic. We shall showthat Rλ (corresponding to the given Eλ) is analytic in Δ except at iso-lated points. Since λ has at most finite multiplicity (by Theorem 3.2)as an eigenvalue of Eλ, it follows that λ 0 Σ'β.

It will be enough to show that (Aλ — X)φ = 0 has a nonzero solutionat only isolated points λ in Δ. For, by Lemma 4.2, Rλ will then existexcept at these isolated points and, by the conditions of the theorem,and Remark 2.2, will be analytic.

Following M. G. Krein (see [1], §84), we introduce an analyticalbasis φx(X)9 , φn(X) for If (λ), λ e π+ U π~ U Δ, by

Φ*(λ) = [1 + (λ - Xo)RMXo) , k = 1, 2, , n .

Here Φi(λ0), , φn(X0) form a basis (for convenience assumed orthonormal)for if (λ0), with λ0 G 7Γ+.

The solution space of (Aλ — X)φ = 0 is i?(λ) Π DΛλ. According toLemma 4.1, this subspace contains a nonzero vector at just those pointsX e Δ which are zeroes in (Δ—) of

(4.1) det <φ,(λ), φ,(λ0) - Fΐφk(X0)> , Xeπ~[J (Δ-) .

As noted, the expression is meaningful also in ττ~~, indeed is analyticthere and continuous in π~ U (Δ—). Thus the theorem can be provedby showing that (4.1), (which is nonvanishing in ττ~), can be continuedanalytically across Δ.

For λ G π+ U (Δ+) we have

F,Φk(X0) = Σ i^(λ)<Mλ0), where Fkι(X) = (Fλφk(X0), φ,(λ0)) .

The coefficient determinant, det (Fkι(X)) is analytic on π+ and continuouson π+ U (Δ + ) . It is non-vanishing wherever F^1 exists, hence in par-ticular on J + .

We shall show that the expression, defined for λ G π+ (J (Δ+),

(4.2) (-1)* det (φz(λ0), Fλφk(XQ)). det <φ,(λ), ψΛ(λ0) - F?φk(X0)>

coincides on Δ with (4.1). Since this expression (4.2) is analytic on π+and continuous on π+ U Δ + , it furnishes the desired continuation of(4.1) across Δ.

Since i*Y+ = ί\*+ on Δ, therefore

= F?+Fλ+φk(λ0) - Fλ+φk(X0)

Noting that Fλ*_ = Fk+, this permits writing the limit value of (4.1) inthe form

(4.3) ( - iy det (Fkl{X +)) det <φ, (λ), φι(\0)

But

= (Fλ+φk(X0), φ,(

, Fλ+φk(XQ))

so that (4.3) is identical with the limit value on Δ of (4.2).The theorem is proved.

5* Strict contractions* In this paragraph we shall examine spectralresolutions of a symmetric operator A satisfying the conditions

( I ) A has equal finite defect numbers (n, n).(II) Every point X on a real interval Δ is of regular type for

A9 i.e. (A — λ)"1 exists and is bounded.Condition II can be stated in the following equivalent form:

(IΓ) Any self-adjoint extension in £%f of A has in Δ only isolatedpoints of its spectrum. No point of Δ is common to the spectra of allsuch extensions.The equivalence of II and IΓ follows from Hartman ([5], prop, (ii))

Let Eμ be a spectral resolution of A, and i*\ be the associatedfamily of contractions of M(X0) into M(λ0). It follows from theorem4.1 that, on any sub-interval of Δ where (a) and (β) hold, the spectrumof Eμ will contain only isolated points.

Our interest here, however, will be in resolutions for which con-dition (γ) of §4 holds on Δ. In this case, by Theorem 4.1 (iii), thespectrum of Eμ includes Δ. We first state a result valid when Δ is theentire real axis &. When (γ) holds on & we shall describe Fλ as afamily of strict contractions.

THEOREM 5.1. Suppose that A satisfies (I), and (II) on &. LetEμ be a resolution of A for which the associated family of contractionsFλ is strict. Then:

The associated minimal self-adjoint extension of A is unitarilyequivalent to the n-fold direct sum of iD with itself, D being the.differential operator d\dx on J 2 ^ ( — oo, oo).

REMARK 5.1. This theorem generalizes results of Coddington andGilbert [4] for ordinary differential operators on a closed bounded inter-

val. Their method of proof appears to be adaptable to handle certainother ordinary differential operators satisfying I and II, in particular,singular operators in WeyΓs limit circle case.

REMARK 5.2. Condition II on & of course implies that A has noeigenvalues. However it is easy to analyze the more general situationin which eigenvalues do occur, provided II on & holds for the restric-tion of A to the manifold orthogonal to the eigenvectors. In that casethe minimal self-adjoint extension is equivalent to the direct sum ofthe discrete part of A with the operator described in Theorem 5.1.

We shall prove Theorem 5.1 as a special case of a more generaltheorem. We now suppose that I, II, and (γ) hold on i By assump-tion, Fλ+, and hence Aλ+, exists for every X e Δ. The assumptions that|| Fλ || < 1 and that A has no eigenvalues on Δ imply that DΛλ+ f] Ί?(λ) ={0} for λ e Δ, and hence that ( 4λ+ — λ)"1 exists. This statement followsfrom the fact, noted by Hartman [5], that when / e ^(λ), for λ e ^ ,is written in the form

/ = f0 + f+ + /-, where f0 e DA, f+ e if (λ0), / - e gf (λ0)

for λ0 e 7Γ+, then | | / + | | = | |/~| | . Then, by assumption I and Lemma4.2, Rλ exists, and is continuous in λ, on π+ U (Δ+).

One may define a basis for gf(λ), X e π+ U Δ, by

(5.1) φfc(λ) =: [1 + (λ - λo)βλ]φ,(λo) , k - 1, 2, , n .

Here λ0 e ττ+, and ^(λo), •• ,Φw(^o) form a basis for §f(λ0). That ψk(X)is in iί(λ) follows from (A* - X)Rλ - 1. That φ^X), , φn{X) are inde-pendent follows from the fact that

1 + (λ - X0)Rλ = {Aλ - λo)(Aλ - λ)-1

has an inverse.1

We shall henceforth identify π+ with the half-plane ^ ( λ ) > 0.The basis (5.1) allows a simple representation for ^Rκ+ = l/2ΐ [i2λ+— Rλ-]:

LEMMA 5.1. Assume that A satisfies I, II, and Fκ satisfies (γ).Then for every X e Δ and every f e

(5.2) ^Rx+f = Σ y*j k iΣ

j,k=i

The matrix Φ(X) is positive definite and continuous in X. Here π+

has been identified with the half plane ^(X) > 0.

1 In what follows only the existence of a continuous basis is needed, not its relation

(5.1) to Rλ.

Proof. Since for every / e 3f?> (A* - X)Rλ+f = /, thereforef G ίf(λ). In terms of an orthonormal basis $19 •••,$* for

gf (λ), ^ # λ + / - J?Ct&, where

Ck =

for some α/rfc G |?(λ). Writing &, , 0n, ψlf , α/rn as linear combi-nations of 0i(λ), •••, ^w(λ) establishes the form of (5.2).

From the known relation

J) ^ 0 for ^ λ > 0

follows

(5.3) (^Rχ+f9f)^0 for

Recalling that i?(λ) is invariant under ^R^+, let {^^i?λ+} denote therestriction of ^Rλ+ to §?(λ). We assert that

(5.4) i[^Rχ.+}Φ, Φ)>® w h e n || || > 0 , ^

In view of (5.3) it is sufficient to show that

(5.5) {^Rλ}ψ = 0 implies ψ — 0.

Suppose that {^R^ψ = 0. Then sf = Rλ+ψ = J?λ_'f belongs to) Π 2>(Aλ«). Writing g in the form

ί/ = 0o + g+ + g~, go

then, by the definition of D(Aλ±),

Since || Fλ+ \\,\\ F λ _ || < 1, this implies that g+ = g- = 0, i.e. βr GSince Rλ+ψ e D(A) therefore ψ e dA(X), the orthogonal complement ofif (λ). Thus ψ = 0. This proves (5.4).

Now let be an arbitrary element of if(λ) and put

In view of the independence of ^(λ), •• ,λn(0), this relation is a one-to-one linear mapping of g^(λ) onto the -dimensional space of vectorsξ = (&, " ' ^ J . Thus relation (5.4) is equivalent, because of the formof (5.2), to

Z<PjkQM£k>0 when | | | | | ^ 0 .

That is, the matrix Φ(λ) is positive-definite.It remains only to prove the continuity of Φ(x). This follows

directly from the relation

since det (0μ(λ), 0, (λ)) =£ 0.

THEOREM 5.2. Suppose that on an interval Δ the operator A satis-fies I, II and that Eμ. is a spectral resolution of A for which the corre-sponding mapping Fλ satisfies (γ). Let A+ in Sίf* he a minimal s.a.extension of A with orthogonal resolution E£ satisfying Eμ = PEμ.For μ e Δ define ρ(μ) = llπ\Φ(μ)dμ. Then the part of A+ on E+{Δ)£έf+

is unitarily equivalent to the multiplication operator on J5f2(p(μ)),μe Δ.

Proof of the Theorems. It is pointed out by Coddington and Gilbert[4] that the multiplication operator in J*f2(p) (where p is strictly increas-ing and continuous in λ on &) is unitarily equivalent to the w-folddirect product of iD with itself, D being the differential operator djdxon jS a(—°°> °°) Thus Theorem 5.1 is a corollary of Theorem 5.2.

For every / e <§ίf and every bounded real interval A9 c A,

(E(Δ')f,f) = limi-

π

Here we have used the continuity of Rκ on π+ \J (d+) and of Eκ on Δ.Let fif(λ) = {gk(X)}l=1 be denned by gk(X) = (/, φk(λ)). Hence

(E(A')f,f) = - U ΣΦ3k(X)gj(X)gh(X)dX7Γ J4'

(5.6) || E+(J')f\\2 = ίJΔ'

Now suppose / e ^ is in E+(Δ)£^+. Thus #(4)/ = /. ConsiderV: E+(Δ')f —> χj'(λ)flr(λ), where χj/(λ) is the characteristic function ofthe interval zf' c J. From (5.6) V is an isometric mapping of Z(Δ)(see Lemma 3.1) into £f*(pfa)), (λ 6 A), which carries i?+(Λ') into theoperation of multiplication by XJf(X). Since Z(Δ) is fundamental on

Theorem 5.2 follows.

6. Differential operators* Let Lu = — (jm')' + ^ be an ordinarydifferential expression on the positive axis 0 x ^ oo, with p and greal measurable functions such that p(x) > 0,

< oo , I I q(χ) \dx <Jo

for any b > 0. With a suitably prescribed2 minimal domain in j£^2(0, oo),L defines a symmetric quasi-differential operator Lo with defect numbers(1,1) or (2, 2). It is easily seen that Lo has no eigenvalues. When thedefect numbers are (2, 2), the Conditions I and II of § 5 are automati-cally satisfied on ^ , so that the results of that section hold.

We shall assume that Lo has defect numbers (1,1), i.e. is in thelimit point case, and shall study the absolutely continuous spectrum ofa minimal self-adjoint extension Li of Lo. As before (§ 3), Li operatesin a space £έfΛ~ containing ^f and has a spectral family of projectionsdenoted by E£.

Let Ma and Ms be the absolutely continuous and singular subspacesof ^f^ with respect to Li (see [7] for definitions). Thus Ma and Ms

reduce Li, are orthogonal, and Sίf^ = Ma © Ms. For any u e Ma[u e Ms],the function (Eμu, u) is absolutely continuous [singular] with respect toLebesque measure on — oo < μ < oo. Let Eμ be a generalized resolutionof Lo for which Eμ — PEμ. By the absolutely continuous spectrum ofLi (or of Eμ) will be meant the spectrum of the part of Li in Ma.The singular spectrum is defined similarly.

It has been proved by N. Aronszajn [2] that the absolutely continu-ous spectrum is the same point set for all orthogonal resolutions of thedifferential operator Lo. The following theorem extends Aronszajn'sresult to generalized resolutions in a way parallel to Theorem 4.1 foressential spectrum. Clause (iii) contains a partial extension, for differ-ential operators, of Theorem 5.1.

THEOREM 6.1. Let Lo be a quasi-differential operator as describedabove and let Σa denote the points of the absolutely continuous spectrumof any (hence every) orthogonal resolution of Lo. // Eμ is an arbitrailychosen (generalized) resolution of Lo with absolutely continuous spec-trum Σf

a and singular spectrum Σf

s, then:( i ) Σ'a^Σa

(ii) When (a) and (β) hold on Δ for the family of contractionsassociated with Eμ, then Σf

a and Σa coincide on Δ.(iii) When (7) holds on Δ, then Δ c Σ'af while Δ Π Σ'8 = 0.

The proof depends upon

LEMMA 6.1. Let p(μ) — [ftfc(/*)];U=i be a nondecreasing Hermitianmatrix (-—00 < μ < 00) and A the multiplication operator with maxi-mal domain in J*f2(P) Let

p(μ) = pa(μ) + ps(μ)

be the Lebesque decomposition of p into its absolutely continuous and

2 A precise specification may be found in [1], Appendix II,

singular parts, defined by the corresponding decomposition of thecomponents of p.

Then pa and p8 are nondecreasing Hermitian matrices. -&ζ(pa)and -S?(ft) are, respectively, the absolutely continuous and singularsubspaces of £%{p) with respect to A. Thus the absolutely continuousspectrum of A consists of the points of increase of pa, or equivalentlyof its trace tr pa. A similar statement holds for the singular spectrum.

We shall omit the proof of Lemma 6.1.

Proof of Theorem 6.1. For J?X Φ 0, let ψλ = ψ(x, X) denote the-Sf solution of Lψ = Xψ which is determined by

[p(x)ψ'(x,X)]x=0 = - 1 .

Put ψ(0, X) = m(λ). Each generalized resolution Eμ of Lo is now specifiedby a family of contractions Fκ: M(i)—>M(—i) of the form

Fλt-< = W(X)ψi

where W(X) is analytic and | W(X) | ^ 1 for ^X > 0. Define

Θ(X) =1 - W(X)

Since ^m(ϊ) > 0 and m(—i) = m(i), therefore ^Θ{X) Ξ> 0 (with θ — oowhen W = 1).

A. V. Straus [11] has associated with each Eμ a spectral matrixPif*) — [Pjk(f*)]j,k=i,2> ~oo<μ<co, which is Hermitian nondecreasing,and such that

"1 C P

tr ρ(μ) = — lim I 7Γ ε-»+0J0

where

(6.D Φ(X) = ":}:rΛ} z : . ^ x > o.

In particular, when ΫF(λ) = 1, ^(λ) reduces to m(λ).Let A be the multiplication operator with maximum domain in

Jέf2(p) and let Li be a minimal self adjoint extension of Lo, with Eμ =PE+. By the reasoning in [4], §4, A is unitarily equivalent to Lo

+.Therefore, by Lemma 6.1, the problem is reduced to a consideration

of the absolutely continuous and singular parts of tr^o. But such con-sideration is possible along the lines of [2].

A set G is a support of a real measure v when v{& — G) — 0. Itis a minimal support when for every support Gx c G, the Lebesque

measure | G — Gλ | = 0. It is easy to prove that when v and 1/ areabsolutely continuous measures with minimal supports G c G ' thenv <v* (i.e. ι/(β) = 0 implies v(s) = 0).

The following disjoint sets Gα and Gs are minimal supports for,respectively, the absolutely continuous and singular parts of t r ^ (com-pare [2]):

Ga = {μ G &\ lim φ(X) exists finitely and lim^φ(X) > 0}

Gs = {μe &\ J^Φ{X) -» 00 when λ -> μ} .

(Here it is understood that X—>u with the constraint that ε < Arg(λ—μ)<π — ε for some fixed ε > 0.)

We shall compare the sets Ga, Gs corresponding to an arbitrarilychosen resolution Eμ. with the special sets G°a, G°s corresponding to theorthogonal resolution for which W = 1. Thus in the definitions ofGlf G°s, φ(X) is replaced by m(λ).

We note first that ^θ, ^φ and ^m are all ^ 0 when ^X > 0.Since lini;^ ^(λ) exists finitely except for λ on a certain set So of

Lebesgue measure zero, inspection of (6.1) and the formula

„ (1 + I θ n^m + (1 + I m\m + θ\2

reveals that GaZD Gl — So. Since these are minimal support it followsthat tr pa >- tr p°a, implying the statement (i) of the theorem.

Next, assume that (a) and (β) hold on Δ. Therefore Θ{X) may becontinued down to Δ with ^θ(μ+) = 0 on Δ. Inverting (6.1) oneobtains the formulas

m =θ - φ \θ - φ\

which show on inspection that Ga Π Δ czG°a. Together with the earlierobtained inclusion, this implies that Ga and G°a coincide on Δ. Sincethese are minimal supports, (ii) follows.

Finally, assume (7) holds on J . In this case Θ(X) may be continueddown to Δ with ^θ{μ + ) > 0 for μ e Δ. Equation (6.1) shows that ^(λ)remains bounded as λ —• μ on Δ and hence that Gs Π Δ — 0. ThusΣf

s n Δ = 0. At the same time, by Theorem 4.1 (iii), Δ does belong tothe spectrum of Eμ, and hence must belong to the absolutely continuousspectrum Σ'a. Q.E.D.

REMARK 6.1. A. V. Straus [12] has shown that when Θ(X) may becontinued to real limit values on the entire real axis—equivalent to theassertion that (a) and (β) hold on &— then E£ has simple spectrum,


This, together with Theorem 6.1 (ii), implies the unitary equivalenceof the absolutely continuous parts of minimal self-adjoint extensionscorresponding to resolutions Eμ. satisfying (a) and (β) on &.

REMARK 6.2. Assume (γ) holds on &. If conditions I and II of§5 hold for Lo then, by Theorem 5.1, the multiplicity of spectrum ofLt will be 1, and the operator equivalent to iD. Simple examplesshow that in general (i.e. without Conditions I and II) the multiplicityof spectrum may well be 2 (the maximum consistent with p being a2 x 2 matrix) and that Lt may even be equivalent to iD 0 iD.

REFERENCES

1. N. I. Achieser and I. M. Glasmann, Theorie der linearen operatoren in Hilbert-Raum,Akademie-Verlag, Berlin (1954).2. N. Aronszajn, On a problem of Weyl in the theory of singular Sturm-Liouville equa-tions, Amer. J. Math., 79 (1957\ 597-608..3. Earl A. Coddington, Generalized resolutions of the identity for symmetric ordinarydifferential operators, Annals of Math., 6 8 (1958), 378-392.4. E. A. Coddington and R. C. Gilbert, Generalized resolvents of ordinary differentialoperators, Trans. Amer. Math. Soc, 9 3 (1959), 216-241.5. P. Hartman, On the essential spectra of symmetric operators in Hilbert space, Amer.J. Math., 75 (1953), 229-240.6. E. Heinz, Zur Theorie der Hermiteschen Operatoren des Hilbertschen Raumes, GottingerNachrichten, Math.-Phys. Kl. Πa, (1951) no. 2.7. S. T. Kuroda, On the existence and unitary property of the scattering operator, IINuovo Cimento, 12 (1959), 431-454.8. M. A. Naimark, Spectral functions of a symmetric operator, Izvest. Akad. Nauk,S. S. S. R. Ser. Mat., 4 (1940), 277-318.9. F. Riesz and B. Sz.-Nagy Functional Analysis, Unger, New York (1955).10. A. V. Straus, Generalized resolvents of symmetric operators, Izvest. Akad. NaukS.S.S.R., Ser. Mat., 18 (1954), 51-86.11. , On spectral functions of differential operators, Izvest, Akad. Nauk. S. S. S.R.,Ser. Mat., 19 (1955), 201-220.12. , On eigenfunction expansion of a second-order boundary problem on a semi-axis, Izvest. Akad. Nauk S. S. S. R. Ser. Mat., 20 (1956), 783-792.13. H. Weyl, ϋeber gewohnliche Differential gleichungen mit Singularitaten, Math. Ann.,6 8 (1910), 220-269.

OPERATORS OF FINITE RANK INA REFLEXIVE BANACH SPACE

A. OLUBUMMO

1. Let X be a reflexive Banach space and F(X) the Banach algebraof all uniform limits of operators of finite rank, in X. Bonsall [1] hascharacterized F{X) as a simple, U*-annihilator algebra: F(X) containsno proper closed two-sided ideals, every proper, closed right (left) idealof F(X) has a nonzero left (right) annihilator, and, given any TeF(X),there exists Γ* e F(X) such that

In this note, we obtain a new characterization for F(X) (Theorem3.2): a Banach algebra A is the algebra F(X) of all uniform limits ofoperators of finite rank in a reflexive Banach space X if and only if Ais a simple, weakly compact, I?*-algebra with minimal ideals (A is weaklycompact if left- and right-multiplications by every ae A are weaklycompact operators). In the process of proving this result, we obtain acharacterization of reflexive Banach spaces which seems to be of someindependent interest (Theorem 2.2): a Banach space X is reflexive if andonly if every operator in X of rank 1 is a weakly compact element ofB(X).

2. Let X be a Banach space and B = B(X) the Banach algebraof all bounded operators in X with the uniform topology. For Te B,let Rτ denote the operator in B obtained by multiplying elements of Bon the right by T: RT(A) = AT for Ae B.

Suppose that T is a fixed operator of rank 1 in X with H =[xe X: Tx = 0]. Then H is a closed hyperplane in Xand if x0 is an ele-ment of X such that Tx0 Φ 0, then X = H 0 (x0) and we may assume that||α>o|| = 1. Write B' - [Si6 B: S(H) - (0)]. For each Se B\ we definean element xs of X by setting xs — S(xQ) The mapping S —> xs is clearlylinear.

LEMMA 2.1. The linear mapping S-+xs is a homeomorphism ofBf onto X.

Proof. It is clear that the mapping is one-to-one and, since||S(ίco)|| ^ | |S | | , it is continuous. It is also onto; in fact, let <£>GX* besuch that φ(H) = (0), φ(x0) = 1. Then for given xe X, the operator Sx

defined by setting Sx(y) = φ(y)x, ye Xbelongs to Bf and is mapped intox by the mapping S—*S(x0). Hence, by the closed graph theorem, the

Received November 9, 1961.

1023

1024 A. OLUBUMMO

mapping is bicontinuous and the proof is complete.Let Bλ denote the unit ball in B, so that Rτ{Bλ) = [PTe B: \\P\\ 1].

LEMMA 2.2. RT(BJ = [ A e B': \\AxQ\\ g || TxQ\\\.

Proof. It is clear that R^B,) a[Ae B':\\Axo\\ ^ \\ Txo\\]. Now leti e B' with HAajoll ^ 112X11; we find PeB, such that A = PT. Thereexists f e X* such that | | ^ | | = 1 and t ( 2 X ) = \\Tχo\\. We define P bysetting Px = ψ(αj)Aa?0/|| 2X11. Then PTα = 0 if # e H and P 2 X = A#o.Thus PT and A coincide in the subspace (x0) and must therefore coincideeverywhere in X. Finally | | P | | = sup,,^,^ I!^(#)A$O||/|| 2X| | ^ 1; hencePeB, and Rτ(Bλ) = [Ae B': | |Aαo | | ^ I |2X| |].

LEMMA 2.3. Lei F be any subset of B'. If FB' denotes the closureof F with respect to the weak topology of Br and FB the closure of Fwith respect to the weak topology of B, then FB' = FB.

Proof. Let P o e FB' and let

N = JV(P0; Φ19 Φ2, , Φn; e)

= [Pe B: \Φk(P - P o ) | < ε; k = 1,2, ^.9n;ΦkeB*]

be an arbitrary neighborhood of Po in B. Then the neighborhood JV' =N(P0; Φ'u Φ'2, --.,Φ'n;e) of Po obtained by taking the restriction of Φk toB' for each k, contains a point P of F. Since Pmust therefore belongto N, it follows that FB' S FB.

Now suppose that PoeFB. Then PoeB' since Br is closed withrespect to the weak topology of B(X) (being linear and strongly closed).Let N' = [Pe B': \ φk(P - Po) | < ε, k = 1, 2, . . , n; φk e (B')*] be an arbi-trary neighborhood of Po in Bf. Then again, by considering theneighborhood N= [Pe B: \ Φk(P- Po) \< e, k = 1,2, . . , n , Φk e β*] obtainedby extending φk to Φk, for each &, on the whole of B, we can findP e ί 7 such that P e JV'. Hence FB S i^B . This completes the proof.

THEOREM 2.1. A Bαnαch space X is reflexive if and only if everyoperator in X of rank 1 is a right weakly compact element of B(X).

Proof. If X is reflexive and T is of rank 1, then by Lemma 2.1,J5' is homeomorphic with X under the correspondence S «-> S(x0). Nowthe image of Bλ under Rτ is a bounded subset of B' which is thereforecontained in a set U which is compact with respect to the weak topologyof Bf and by Lemma 2.3, with respect to the weak topology of B(X).Thus Rτ is a weakly compact operator in B(X) and T is a right weaklycompact element of B(X).

OPERATORS OF FINITE RANK IN A REFLEXIVE BANACH SPACE 1025

Now suppose that Rτ is weakly compact in B(X). Then RT{B^ iscontained in a set V c Bf which is compact with respect to the weaktopology of B{X) and hence also with respect to the weak topologyof Bf. Now the ball Q = [Ae Br: \\A\\ | | Txo\\l\\xo\\] is contained in

RT{B^) c V and is weakly closed. Hence Q is compact with respect tothe weak' topology of Br and therefore Bf is reflexive. Since Br ishomeomorphic with X, it follows that X is reflexive and the proof iscomplete.

COROLLARY 2.1. If X is a reflexive Banach space, then the algebraF(X) of all uniform limits of operators of finite rank in X is aweakly compact algebra.

COROLLARY 2.2. (Ogasawara [2] Theorem 4.) Let H be a Hilbertspace and B(H) the Banach algebra of all bounded operators in H.If T is a compact operator in H, then T is a weakly compact elementof B{H).

3. This section is devoted to the study of simple, weakly compact,i?*-algebras with minimal ideals.

LEMMA 3.1. Let Abe a simple Banach algebra with minimal ideals.Then every maximal regular left ideal M of A has a nonzero rightannihilator.

Proof. Since A is a simple Banach algebra, there exists anidempotent ee A such that M Π Ae = (0) and i l ί φ i e = i . Since Mis regular, there is je A such tha't xj — xe Mfor every xe A. For someα0e A and m 0 e l , j = m0 + αoe, aoe Φ 0. Suppose now that m is anarbitrary element in M. We have mj — me Mand mj — maoe = mm0e M,from which it follows that m — mαoee M. Now, me M and hencemaoe e M. However, maoe e Ae since Ae is a left ideal, thus maQe e Mf] Ae =(0) and since m is arbitrary in M, the lemma is proved.

LEMMA 3.2. Let A be a simple Banach algebra with minimal rightideals. IfjeA and j has no left reverse, then there exists a Φ 0 suchthat ja — a.

Proof. Let J = [yj — y: ye A]. Then J is a regular left ideal ofA which is proper since j $ J. Hence by Lemma 3.1, there exists aeA,a Φ 0 such that Ja = (0), i.e. such that yja — ya = 0 for all ye A orA(ja — a) = (0). Since (A)r = (0), this implies that ja — a.

LEMMA 3.3. Let A be a simple B*-algebra with minimal right

1026 A. OLUBUMMO

ideals. If \ \ is any other norm in A with \a\ S \\a\\ for each α e A,

then I I = || | | .

Proof. Lemma 3.2 implies t h a t if | | is any other norm in A, then

lim^oo \an\lln = lim^oo \\an\\l!n for every α e A (Cf [4], Lemma 3.1). Then

since A is a J3*-algebra, we have

| α # | | α | ^ |α*α| ^ lim | ( α % ) w | 1 / w

and since |α*| ^ ||α*|| and |α | ^ | |α | | , the result follows.

THEOREM 3.1. A Banach algebra A is the algebra F(X) of alluniform limits of operators of finite rank in a reflexive Banach spaceX if and only if A is a simple, weakly compact, B%-algebra withminimal right ideals.

Proof. Let A be a simple, weakly compact, J5*-algebra with eA aminimal right ideal, e a primitive idempotent. We represent A as analgebra of operators S$f in eA, the latter regarded as a Banach space.Corresponding to each a e A, we define an operator a e j y by a: x ~> xafor x e eA. The correspondence a —> a is obviously an isomorphism andif we take | |α | | = supMβnSi | |αα||, xe eA, the correspondence is an isometryin view of Lemma 3.3. Thus A is isomorphic and isometric to jzf andA is the uniform closure of

Next we show that eA is a reflexive Banach space. Now e has noleft reverse in A; hence by Lemma 3.2, there exists ae A, a Φ 0 suchthat ea — a. The set P = [αe A: eα = α] is a right ideal of A and sinceP ξ^ eA, we must have P — eA since eA is minimal. If e is now regardedas a left weakly compact operator on A, then it is clear that the setP = eA is a reflexive Banach space.

Our next step is to show that in the representation described above,j ^ contains all operators of finite rank in eA. Corresponding to eachaeAe, there exists a continuous linear functional <pa on eA satisfying<pa(x)e — xa, xe eA. Let G = [<pae (eA)*: ae A]; then G is a linear

subspace of (eA)*. We show that G is closed with respect to the usualnorm in (eA)* defined by \\g>\\ — sup^n^ \<p(x)\ %e eA. For aeAe, wehave xa — φa(x)e, xeeA, and since | |α | | = | |α | | for each aeA, we have

| |α | | = | |α | | = sup \\xa\\ aeAe

= sup \\φa(x)e\\11*11^1

= sup |?>α(aOIIM|l l l

OPERATORS OF FINITE RANK IN A REFLEXIVE BANACH SPACE 1027

Thus G is topologically equivalent to Ae and hence closed. Having provedthat G is a closed linear subspace of (eA)*, we now show that G is infact the whole of (eA)*. Suppose that there exists φ'e(eA)* such thatφr$G. Since G is closed, there exists Φe(eA)** such that Φ(φa) = 0for all φaeG and Φ(φf) — 1. However, eA is a reflexive Banach space:hence there exists uoe eA, uQ Φ 0 such that Φ(ψ) — <p(uQ) for all φe (eA)*.In particular, for φa e G, this implies that 0 = φa(uύ)e — uoa for all a e Ae,which in turn implies that uQe (Ae)L = (0) which is absurd. Hence G =(eA)*. From this it follows that sf contains all operators of rank 1and hence all operators of finite rank in eA, since if T is an operatorof rank 1 in eA, then there exists φe (eA)* and uoe eA such that xT =φ(x)u0, xe eA. Since φeG, there exist ae Ae and φae (eA)* such that<P = φa and #α = <pα(x)e. Let uQ = βα0 for some αoe L; we have #Γ =φa(x)u0 = φa(x)ea0 =xaa0, and since aaoe A, the operator aaQ — T belongsto j y .

Finally, the uniform closure of the set of all operators of finite rankin eA is a closed two-sided ideal of sf which must coincide with s/ sincej ^ is simple. Thus the "if" part of the theorem is proved.

That F(X) is a simple, weakly compact jB*-algebra with minimalideals follows form corollary 1 and a result due to Bonsall and Goldie[1], Theorem 2. This completes the proof of the theorem.

REMARKS. 1. The problems discussed here were suggested byreading Ogasawara and Yoshinaga [2,3] and Bonsall [1].

2. Work on this paper was started at University College, Ibadan,and completed at Yale University. The author wishes to express hisgratitude to the Carnegie Corporation of New York and to Yale Universityfor financial support.

REFERENCES

1. F. F. Bonsall, A minimal property of the norm in some Banach algebras, JournalLondon Math. Soc, 29 (1954), 156-164.2. T. Ogasawara, Finite dimensionality of certain Banach algebras, Journal of Science,Hiroshima University Series A 17 (1953), 359-364.3. T. Ogasawara and K. Yoshinaga, Weakly completely continuous Banach *-Algebras,Journal of Science, Hiroshima University Series A 18 (1954), 15-36.4. A. Olubummo, Left completely continuous B^ algebars, Journal London Math. Soc, 32(1957), 270-276.

UNIVERSITY COLLEGE, IBADAN, NIGERIA

AND YALE UNIVERSITY

ON THE APPROXIMATION OF FUNCTION SPACESIN THE CALCULUS OF VARIATIONS

DAVID A. POPE

Introduction^ A basic feature of most of the methods used for thenumerical calculation of a variational problem is the reduction of theinfinite dimensional problem to a finite dimensional problem by somekind of approximation. One of the most natural approximations is thatof replacing a curve or a surface by a finite number of points lying onor near the curve or surface. The points are then connected by simplearcs or surfaces, and the resulting approximation will, if the number ofpoints is sufficiently large, presumably be close to the original curve orsurface. The difficulties inherent in this approach to surface problemsare well illustrated in the works of Rado [8], [9] on surface area.

The replacement of a curve by an approximating polygon, however,does lead to a usable finite dimensional approximation scheme. Lewy[3] (Chapter IV) gives a proof of the existence of an absolute minimumto the positive regular nonparametric problem by using such an approxi-mation scheme, and his proof could be used to design a numerical processfor approximating this minimum.

The methods of algebraic topology which M. Morse ([4] to [7]) appliedto the calculus of variations have led to a greater understanding of therelationships between all the extremals to a variational problem. Theextremals are classified according to their index types, in analogy withquadratic forms of a finite number of variables. While the extremalswith nonzero index are not minimizing, they are of importance in manyphysical applications.

In this paper we shall treat the problem of computing the non-minimizing extremals as well as those of minimizing type, using thetheory developed by Morse, together with a general theory of approxi-mation. In part 1, a brief restatement of some of the principal defini-tions and theorems of Morse [6] will be given, in the current languageof algebraic topology. In part 2, a general theory of approximation toan abstract metric space will be developed, and the convergence of theapproximations to the critical levels of the problem defined on this spacewill be demonstrated. Part 3 will show that the polygonal approxima-tions to curves leads, in the parametric problem, to approximationssatisfying the theory of part 2.

The structure of part 2 is given with sufficient abstraction so that

Received July 24, 1961. The preparation of this paper was sponsored by the Office ofOrdnance Research, U. S. Army. Reproduction in whole or in part is peimitted for anypurpose of the United States Government.

1029

1030 DAVID A. POPE

it may be applied to any reasonable method of approximation to a varia-tional problem; thus we are not necessarily restricted to the polygonalapproximations described in part 3.

1. Outline of the Morse theory • Given a metric space M, and areal valued function F defined on M, we define the sets Fa as the setof all points x of M such that F(x)^a. We make the basic assumptionof bounded compactness; that is, the sets Fa are compact for all a.

We now assign a homology theory to M and its compact subsets.The most generally useful homology for the calculus of variations hasbeen the Cech theory, in the form given by Vietoris (see Vietoris [11]and Morse [6]). In this paper we shall use Cech homology, as definedin Eilenberg and Steenrod [1], Chapter IX, with coefficient group thefield of integers modulo 2. We shall largely follow the notation ofEilenberg and Steenrod [1].

For a ^ β > 0, we define the inclusion map

iί (Fay Fa-ct) > (Fa, Fa_β)

of the compact pairs. Then for a ^ β > 0 and for each q we have thehomomorphisms

τr£: Hq(Fa, Fa-Λ) > Hq(Fa, Fa_β)

induced by ig, with π;=identity, and π\πi—πl for a Ξ> /3:> 7.This set of groups and homomophisms defines a direct system (Eilen-

berg and Steenrod [1] p. 212). We may then take the direct limit

dir lim Hq(Fa9 Fa^) = Hq(Fa, FaJ)

and define this as the cap group of index q at the level F = a. Morse[6] defines an equivalent group of cap classes in a different way, usingthe Vietoris homology theory.

If Hq(FO9 FaJ) Φ 0, we will say that F — a is a critical level of Fon M, with index q. Now we derive three lemmas about the cap groupswhich will prove useful.

LEMMA 1.1, Given any nonzero element V of Ha{Fa, FaJ)y for everysufficiently small a > 0, there exists a nonzero element V* in Hq(Fa, Fa-a)such that the projection

πa: HQ(Fa, Fa^a) > Hq(Fa, FaJ)

maps VΛ into V.

Proof: By Lemma 4.3, p. 221 of Eilenberg and Steenrod [1], thereexists a positive number 7 and an element Vy in Hq{Fa, Fa-y) such that

APPROXIMATION OF FUNCTION SPACES 1031

πy Vy = V. Now choose a arbitrarily between 0 and 7, and set VΛ = πl Vy.Clearly Va satisfies the lemma, since πΛVa = πΛπlVy — πyVy — V.

LEMMA 1.2. Suppose there is no critical level F — c of index q inthe half-open interval b < c <£ α. Then Hq(Fa> Fb) = 0.

Proof. We shall prove this lemma by contradiction. SupposeHq{Fa, Fh) Φ 0, and let U be a nonzero element of this group. We de-note the homomorphisms of Hq(Fa, Fa) into Hq(Fa, Fβ) induced by inclu-sion, for a < β < α, by jg. These are the projections of the direct systemdefining the cap group at the level F — a. This cap group is zero, sinceF = a is not a critical level of index q.

Next we define s as the supremum of all numbers β in the interval[6, α] such that U is woέ in the kernel of jξ. By Lemma 4.4 of Eilen-berg and Steenrod [1], p. 221, there is a 7 < a such that jy

bU = 0, sincethe direct limit is zero. Hence for all β with 7 ^ β ^ a, jξU = jξjy

bU =0. Therefore by definition, s ^ 7 < α.

Now suppose /5 < s and jζU = 0. Then for every 7 with β < 7,3lU = ijϋj£Ϊ7 = 0, and /S is an upper bound, contrary to the definitionof s. Hence for all β < s,jβ

bUφ 0.Since, by definition of s, if U — 0 for all β > s, the map j'J satisfies

the equation

j Z7 = inverse limit jζU — 0β>s

by Lemma 3.11, p. 218 of Eilenberg and Steenrod [1]. Now consider thefollowing portion of the exact sequence of the triple (Fb, Fs, Fa)

Hq{Fs, Fb) > Hq(Fa, Fb) > Hq{Fa, Fs) .

Since U is in the kernel of j'bf it is in the image of i. Hence there isa nonzero element V in Hq{Fs,Fb) such that iV= U.

But F = s is not a critical level of index q; hence the direct limit

as β~< s of Hq{Fs, Fβ) = 0. Then by Lemma 4.4 of Eilenberg and Steen-

rod [1], p. 221, there is a 7 < s such that V is in the kernel of

j:Hq(Fs,Fb) >Hq(Fs,Fy).

Consider now the following portion of the homomorphism of the exactsequences of the triples (Fa, Fs, Fb) and {Fa, Fsy Fy) induced by inclusion:

Hq(F$, Fb) > Hq(Fa, Fb)

1032 DAVID A. POPE

The element V in Hq(F8fFb) satisfies iV = U, and so jbiV — iyjV =iy0 = 0. Hence U is in the kernel of j \ and Ύ < s. This contradictionproves there is no nonzero element U in Hq(Fa, Fb), thus proving thelemma.

LEMMA 1.3. Suppose there are no critical levels F = c of index qor q + 1 on the half-open interval b < c ^ a. T/ ew ί/ β inclusion mapof Fh into Fa induces an isomorphism of Hq{Fh) onto Hq(Fa).

Proof. By Lemma 1.2, Hq(Fa, Fb) = Hq+1(Fa, Fb) = 0. Hence in theexact sequence of the pair (Fa, Fb) we have

0 - ? - Hq(Fb) - U Hq(Fa) -U 0 .

Therefore i is an isomorphism, and the lemma follows.

2. Approximations to a metric space* The following problem isnow defined: we are given a metric space M of points x and a real valuedfunction F defined on M; we wish to find the critical values of F on M.To do this we define a sequence of approximations to the space M. Usingthe methods of algebraic topology which M. Morse ([4] to [7]) appliedto the calculus of variations, we are able to measure how close to thecritical levels of the original problem those of the approximated problemwill lie.

Following Morse [6], pp. 29-36, we repeat here some definitions forthe convenience of the reader. An admissible deformation of a subsetE of M is defined as a homotopy q(p, t): E x I —> M, where / is the inter-val 0 ^ t ^ T, and q(p, 0) = p for all p in E. The curve q(p, t) obtainedby holding p constant is called the trajectory defined by p. If the pointsri — Q(P> *I) and r2 = q{p, t7) are on the trajectory defined by p with0 ^ tx ^ t2 ^ T, then rx is said to be an antecedent of r2.

The admissible deformation is said to admit a displacement functiond(e) on E if, whenever rx is an antecedent of r2 with the distance fromn to r2 greater than ε > 0, then F(r^) — F(r2) > δ(ε), where d(ε) is apositive single valued function of ε. If an admissible deformation of Eadmits a displacement function on each compact subset of E, the defor-mation is called an F-deformation.

The function F is called upper-reducible at p if for each constantc > F(p), there exists a neighborhood of p relative to Fc which posessesan JP-deformation carrying the neighborhood into a set lying in Fc-e forsome positive e.

Following Morse, we make the assumptions that the sets Fa arecompact for all α, and that F is upper-reducible at all points of M.Under these assumptions Morse ([6], p. 38) proves that each critical


level ("cap limit") contains at least one homotopic critical point.Next we give a set of formal requirements defining a set of approxi-

mations to the space M which are admissible with respect to F. To dothis we define a sequence {pn} of functions, called approximations, withthe following properties:

(1) For each n, pn is a continuous function of M into M. The imageof M under pn will be called Mn.

(2) Mn is a closed subset of M for each n.(3) F is a continuous function on M* for each n.(4) For any real number α, and any e > 0, there is an integer N

such that

(2.1) F(pnx) g F(x) + e

for all n > N, and for all x in Fa.

(5) For any α, β > 0, n > N of property 4, the composite map

(2.2) ip n : Fa > Fa+e Π Mn > Fa+e

is homotopic to the inclusion map

(2.3) j : Fa > Fa+e .

The map pn in (2.2) is regarded as taking points of M into pointsof Mn, where Mn is regarded as a separate space, with topology inducedby the topology of Λf. The inclusion map i of (2.2) is the map takingpoints of Mn into themselves, considering Mn as a subset of M in theimage. We shall abbreviate by setting

(2.4) Fh

n = Fbf]Mn .

For the applications of this theory to computing variational problems,we shall demand one further restriction on the subspaces Mn; they areto be finite dimensional of dimension rny in the sense that they are locallyhomeomorphic to euclidean space of rn dimensions. In this manner theproblem of finding the critical points and levels of the infinite dimensionalspace M is reduced to the simpler problem of finding the correspondingobjects in the finite dimensional subspace Mn. However, as this restric-tion is not used in the topological arguments to follow, it is not placedin the set of requirements above.

First, the properties of bounded compactness and upper reducibilitymust be verified for the subspace Mn.

LEMMA 2.1. Under the conditions (2) and (3) of 2.1, the subsets Fb

n

are compact, and F is upper reducible on Mn.

Proof. Mn is closed by requirement (2), hence Fb

n is compact. F

1034 DAVID A. POPE

is upper reducible on Mn because any continuous function on a space isupper reducible (Morse [6] p. 37).

Lemma 2.1 allows us now to use the Cech homology theory withthe compact subsets F* and the theory developed by Morse to indicatethe relationship between the critical value of F on M and those of Fon Mn. The theorem below gives this relationship in one case of interest,and show convergence of the critical levels on Mn to those on M, asn—> co.

THEOREM 2.1. Suppose F — c is a critical level of index q of F onM, and there are no other critical levels of F on M of index q or q + 1on the interval [c, c + h] for some h>0. Suppose futher that there areonly a finite number of critical levels of F on Mn, for each n.

Then for every sufficiently small e > 0, there is an integer N suchthat for n > N, the approximating space Mn possesses a critical levelcn of index q, with c cn fg c + e.

Proof. Since F = c is a critical level, the cap group Hq(Fc, FCJ) isnontrivial. Let V be a nonzero element of this group. Then by Lemma1.1, for every sufficiently small a > 0, there is a nonzero element Va inHQ(FC, Fc-V) such that the projection of this group into the cap groupmaps VΛ into V. Now choose such a number a < h, and call it a0. Nextpick arbitrarily a positive number e < a0. Finally choose a positive numbera < e. Then by property (4) of the approximations we may choose Nso that for n > N we have

pn(Fc) c F β + i c Fc+e

andpn(Fc-a) c Fc

n^+e c Fc,

The composite map ipn as defined in equation (2.2) then defines a mapof the pairs (FC1 Fc-a) into (Fc-e, Fc-a+e), which is homotopic to theinclusion map of these pairs, by property (5) of the approximations.Therefore the homomorphism

i*p*: Hq(FCf Fc-Λ) > Hg(Fc+e, Fe^+e)

induced by this pair map is the same as the inclusion homomorphism forthe pairs, by the homotopy axiom of homology theory.

Now consider the following portion of the map of the exact sequencesof the pairs (Fc, Fc-a) and (Fc+e, Fc_Λ+e) induced by inclusion:

ά > Hq(Fc) > Hq(Fc, F

Hq(Fc-a+e) > Hq(Fc+e) > Hq(Fcbey


Since there are no critical levels on M of index q or q + 1 in theintervals (c — a, c — a + e] and (c, c + e], Lemma 1.3 allows us to con-clude that the inclusion homomorphisms if and i} are isomorphisms.Then the "five" lemma shows that i3* is also an isomorphism. But it =i*£>*. Therefore the kernel of p* must be zero.

This means that Hg(F?+9, F?_ω+e) contains the nonzero element pt VΛ.Therefore by Lemma 1.2, there is at least one critical level cn of F onMn of index q, in the interval (c ~ a + e, c + e]. Now suppose thereare no critical levels of index q of F on M* in the interval [c, c + e].Then there must be a critical level cn below c, in the interval (c — a + e,c). But a may be chosen so that c — a + e is arbitrarily close to c.This is clearly contradictory since there are only a finite number ofcritical levels on jfcf , by hypothesis. Therefore there is at least onecritical level cn in the interval [c, c + e], and the theorem is proved.

Next we show convergence of the critical points on Mn to those onM in a simple case.

THEOREM 2.2. Given the conditions of Theorem 2.1 for q = 09 sup-pose the function F attains its absolute minimum on M at the pointx, and this minimum point is unique.

Then the approximating spaces Mn contain homotopic critical pointsof index zero, and a subsequence of these points converges to the pointx.

Proof. Let F(z) — c. This level is clearly a critical level of indexzero. Then by Theorem 2.1, the approximating spaces Mn contain criticallevels cn of index zero, which approach the level c from above. ButMorse [6] p. 38 proves that each critical level cn contains at least onehomotopic critical point of index zero. Choosing one such point for eachn, we denote it xn. Since the infinite set {xn} is contained in the compactset Fc+e for some e > 0, it has at least one accumulation point, whichwe may call y, and a subsequence converges to y. But the lower semi-continuity of F implies F{y) ^ \\mnF(xn) = c, for any convergent sub-sequence. Clearly the inequality is impossible, since c is the absoluteminimum of F on M. Hence we have F(y) = c, and therefore y ~ x,since this minimum is unique.

3Φ The approximation of the parametric problem with fixed endpoints. In this part the requirements of § 2 will be applied to approxi-mations to a general class of fixed end point problems in parametricform. The definition of the parametric problem will follow closely thatgiven in Morse [6].

3.1. The curve space. The space M of § 2 will in this application

1036 DAVID A. POPE

be replaced by the space Ω of all continuous curves (parametrized curveclasses) between two fixed points a and b on a compact subset of euclideanw-space, which we shall denote by Σ.

A parametrized curve, or p-curve, on Σ is defined as any continuousfunction from a real interval into the space Σ.

Suppose two p-curves ηx and % are given in the form

Vϊ Q =

%: Q = ?2(ί) 0 £ t £ d .

Let w be any sense-preserving homeomorphism between [0, c] and [0, d]9

and let d(w) be the maximum distance between the points qλ{t) and q2(wt)for έ in [0, c]. The Frechet distance between rj1 and % is defined as

taken over all possible homeomorphisms w. (Cf. Frechet [2].) The setof all p-curves at zero distance from a given p-curve will be called acurve class, or simply a curve. A p-curve belonging to a curve classwill be called a representation or parametrization of that curve.

A p-curve joining a to b on Σ is defined as a p-curve

ψ q = gr(ί) 0 ^ ί ^ C

with the property that

#(0) = a and q{c) = b .

Clearly every p-curve in the curve class of η has this property. Thecurves representable by p-curves with this property are the points ofthe space Ω. The Frechet distance between two curves a and β of Ωis defined as the Frechet distance between any parametrization of a andany parametrization of β. This distance is clearly independent of thechoice of the parametrizations.

3*2. μ-lengtbu A special parametrization of curves given the name/i-length by Morse has been used in many connections in the calculusof variations. A summary of some of its important properties will bemade here.

To define the μ-length of a curve η, we take any parametrizationq = q(t), 0 2g t ^ c, of 7j, and pick a set {ί,-} of k values of t with 0 <Ξίx < *2 < it ^ c- This set defines a partition P = {g,j of the curve37, where #,• = q(tj). We denote the minimum of the fc — 1 distances(?i, Q3+1) on 21 for i = 1, 2, •••, fc — 1, by m(P). Then the sup ra(P)taken over all such partitions with k points will be called μk. We thenset

(3.2) μv = £ - | ^ .

μv is the /^-length of η, and is independent of the particular choice ofthe parametrization q(t) of η. Thus the /^-length μ{τ) of the part of q(t)from t <Z 0 to t g τ may be used as a parametrization for η. This para-metrization has the following properties, proved in Morse [5], and listedalso in Morse [6] p. 34-35.

(1) If q = q(t) is any parametrization of a curve ?], then the μ-parametrization of rj has the form

(3.3) η:q = q(μ) = q(t(μ)) 0 ^ u ^ u,

where t(μ) is a continuous nondecreasing function of μ on the closedinterval [0, μη],

(2) The value of μ at any point q on the curve η satisfies the ine-quality

(3.4) <L ^ μ <ς d

where d is the diameter of the set of points preceding q on η.(3) The μ-length μη of a curve η is a continous function of η on

the curve space Ω.(4) For a given η, the /^-parametrization of

is constant with respect to μ on no subinterval of [0, μv].(5) The parametrization q(μ; η) of a curve η of Ω is a continuous

function of μ and ^ for μ in [0, μ j and η in β.Suppose η is a straight line of length s joining two points p, q in

euclidean space. For any partition Pk with k values {qj} on η, m(Pk) isclearly g s/k — 1. But the equidistant partition gives m = s/k — 1.Hence μk = s/k — 1, and the ^-length /je, of 07 is

If Ύ] is any rectifiable curve joining p to q with are length s, m(Pk)is still S s/k — 1 for any partition Pfc of A; points. Hence μk ^ s/fc — 1,and therefore

(3.6) / ^ s l o g 2 .

3*3 The functional i 7. Having defined the curve space Ω, we nowconstruct the functional F on Ω, We are given a function /(αsj, , %n,

1038 DAVID A. POPE

Pi rn) = f(%, r) of 2n variables with the following properties (n is thedimension of 2*):

(A) f(x, r) is of class C4 in (x, r) for x in Σ, and any set of numbers(r) Φ (0).

(B) / is an invariant under the transformations of local coordinates(cf. Morse [6] p. 64-65).

(C) f(x, r) > 0 at every point of Σ and for all r.(D) / is positive homogeneous of degree 1 in r.(E) The rank of the determinant

is n — 1, and all its characteristic roots except the zero root are positive.Assuming that Σ is arc wise connected, we define a secondary metric

[<Zi?a] on points of Σ as follows. For any two points q19 q2 of Σ we con-sider the class of all rectifiable curves on Σ joining qx to q2; the integral

(3.7) F= [Q2f(x,x')ds

is computed over this class, and the inf F over this class of curves is

[<M2].To define F on an arbitrary curve η of Ω we form the sum

(3.8) Σ

where the points qt are partition points on rj. The sup of s over allpartitions of η is defined as F{η). F(ή) is equal to the integral of / alongthe curve TJ if η is rectifiable, and is infinite if rj is not. Morse [6] showsthat under these conditions, we have bounded compactness of M9 andupper reducibility of F.

3A. Compactness and rectifiabUity Before showing properties 1 — 5of § 2 are satisfied in the parametric problem, we need the followingcompactness lemma:

LEMMA 3.1. If Γ is a compact subset of the curve space Ω, theset A of all the points of Σ which lie on curves of Γ is a compactsubset of Σ.

Proof. (Cf. Morse [6] p. 59.) Let {qj} be an infinite set of pointsof A. Each point qs lies on at least one curve j3- of Γ. Pick one such7y corresponding to each qd and consider the sequence {jj} of curves. Weparametrize each curve of Γ with the μ-length defined in 3.2, and there-fore we have a unique number μά defined as the μ-value of the point


q3 on the curve y3. Since Γ is compact, the curves {7,-} possess a limitcurve To and the //-length μΊ of the curves of Γ is bounded above; hencethe sequence {μ3} possesses a limit point μ0.

Consider the point q0 = 70(/Ό) Qo is clearly a limit point of thesequence {q3}. Hence the lemma follows.

Now let Σa be the set of all points of Σ which lie on curves of Fa;the compactness of Fa then implies the compactness of Σa by the previouslemma. Therefore the set Ta of (x, r) space

(3.9) Ta:xinΣa, Σr\ = l

is also compact.

Therefore the function f(x, r) is bounded above and below by M and

m > 0 respectively for x, re Ta. Thus we have for any p, qe Σ

fds ^ M(pq)

V

for any rectifiable curve joining p to q on Σ. Therefore

m(pq) ^ [pq] ^ M(pq) .

Therefore on any partition {q3} of a curve r] on Ja we have

mΣ(q3qj+1) ^ Σ[q3qj+1] ^ MΣ(q3q3+1) .

Taking the limit for norm of partitions —> 0, we find

(3.10) mh(rj) g F(η) ^ ML{η) .

Thus L(^) ^ j(v)lm ^ φ , where L()7) is the arc length of η.Therefore we have shown

LEMMA 3.2. The curves of Fa are rectifiable and of arc lengthg α/m, where m > 0 is the minimum of f{x, r) on Ta.

3 5. Definition of the approximations. Given any curve of Ω, wemay parametrize it in terms of the μ-length described in 3.2. If thisis done we denote the curve rj by

(3.11) y:q = q(β; η) 0 ^ μ ^ μv .

Making the linear substitution

(3.12) t = μ/μv

we have a new parametrization of η:

(3.13) y:q = q(t) - q(tμv; η) .

1040 DAVID A. POPE

This parametrization of the curves of Ω will be called the uniformί-parametrization.

Now to define the approximations pn, we take a sequence of parti-tions Pn of the unit interval as follows: Pn will be a set of n + 2 points{£;} j — 0, 1, , n + 1, with t0 = 0 and ίΛ+1 = 1, and ΐ i + 1 > t3 . Thenorm of the partition Pn will be denoted δn. We require further thatδn —» 0 (as n—• oo).

We now take a curve r] oί Ω, parametrized as in equation (3.13)and define the points

(3.14) qs = q(td) = q(tau^ η) .

The points qό all lie on the curve η, and q0 = α, qn+1 = 6, the end pointso f Ύj.

Lemma 3.1 shows that the set of ^-lengths of the curves in Fa isbounded. Suppose μv < M for all η in Fa. Then

(3.15) Δμ5 - (tj+1 - ί , K < δnM

for all curves of Fa. Therefore the diameter dό of the subarc of η fromqj to qj+1 satifies

(3.16) di SL 24μj < 2δnM .

Under the conditions described in 3.3, there is a fundamental distancep in the compact set Σa (described in § 3.4) with the property that ifp, q have distance (pq) < p, there exists a unique extremal arc joiningp to q which gives to F a proper minimum value over the class of allarcs joining p to q, and this extremal arc is a member of a field ofextremals covering the ^-neighborhood of the point p simply (except atp). Hereafter, when dealing with a set Fa9 we shall assume that n islarge enough so that the diameters dj of the subarcs of η are all lessthan p.

We then construct the polygon through the points qOf qly , qn+1.The arc of the polygon from q3- to qί+1 is defined as the euclidean straightline from q3- to q3 +1. This polygon obtained from the curve Ύ] is denotedby Pn(V)> a n d is a continuous parametrized curve class, which may alsobe denoted by pn(y), is in Ω. The space pn(Ω) of all polygons obtainedfrom the partition Pn will be called Ωn.

3.6* Verification of the approximation requirements* Having de-fined the sequence {pn} of approximations to the parametric problem, wenow seek to show that they satisfy the topological requirements 1-5of § 2. The following lemmas dispose of properties 1-3.

LEMMA 3.3. For each n, the approximation pn is a continuous

function of the curve space Ω into itself.

Proof. The /^-length μv of η is a continuous function of η in Ω;and the points q3- = q{t3), where q(t) is the uniform t parametrization off] are therefore continuous functions of Ύ) in Ω. This means, given anye > 0, there is a δ > 0 such that d(^, £) < δ in β implies the distance(flit ro) < β in I7 for all j , where q3, r3 are the points on η, ζ respectivelywith ί-value t3. But if the distance between the corners q3 and r3 ofthe two polygons is less than e, the Frechet distance in Ω between thepolygons is also < e. Hence the lemma follows.

LEMMA 3.4. Ωk is a closed subset of Ω.

Proof. The limit of any sequence of polygons with k corners canbe only a polygon with k or fewer corners, hence contained in Ωk. ThusΩk is evidently closed.

LEMMA 3.5. F is a continuous function on Ωk.

Proof. By property 3.2, given any curve a in Ωk and any δ > 0,there is a p > 0 such that β in Ωk, d(a, β) < p implies \μΛ — μβ\ < δ.Then equation 3.5 of § 2 implies that the arc lengths of a and β differby less than δ log 2, since they consists of straight line segments. ButTonelli ([10] vol. 1, p. 304) proves the following theorem:

Given any curve η, and any ε > 0, there exist two numbers δ > Cand p > 0 such that if

d(V, ξ)> P

and

\L{η) - L(ξ)\ < δ (L(η) - arc length of η)

then

\F(V)~F(ζ)\<ε.

Hence Lemma 3.5 follows immediately.

In order to prove requirements 4 and 5 of §1.1 we shall use thefollowing lemmas.

LEMMA 3.6. \impnη = η as n—• oo, uniformly for η in Fa.

Proof. Equation (3.16) of § 3.5 states:

dj ^ 2δηM

whenever η is in Fa. Hence the distance from a point on the straight

1042 DAVID A. POPE

line from q3- to qj+1 to any point of r] between the same points is closerthan 2dj, so the Frechet distance between rj and pj] is less than &dnM.But the dn approach zero, hence the lemma follows.

Now we show the uniform convergence of the arc length of pj] tothat of the broken extremal associated with pj] for η in Fa.

Consider the set of all extremal arcs for the problem, parametrizedby arc length s, joining the points p and q on Σ, and satisfying (pq) < p,the elementary length defined in 3.5. These extremals satisfy the Eulerdifferential equations

(3.17) £t[Fri(gfg')] = Fmι(g,g')

which may be written in the form

(3.18) g'/ = φfa, g')

where φ is the function obtained by the solution of the implicit equations:

F σ' 4- F a" — F = 0

(3.19) * ' j j (summed on i) .g\g\ - 1 = 0

Under the assumptions made on F, the equations (3.19) can be solveduniquely for φi9 and φ{ is a continuous function of g and gf. Since (#, gf)lies in the compact set Ta, \ φ{ \ is bounded. Thus for all the geodesiesconsidered,

(3.20) \gl'(s)\<M.

A broken extremal consisting of the unique elementary extremalarcs joining the points q3t qί+1 of an approximation pn{η) will be calledthe broken extremal associated with pjy])-

Under these conditions, we can prove the following lemma.

LEMMA 3.7. The arc length of the polygon pn(η) approaches thearc length of the broken extremal associated with PJjf) as n—+ oo uni-formly for Ύ) in Fa.

Proof. Consider the pair of points qjf qj+1 of Σ. We shall comparethe arc length of the extremal

®% = 0i(s) 0 ^ s ^ Si

(parametrized by arc length) with the length of the straight line fromq3- to qj+1. Consider the family of straight lines drawn form the initialpoint q3- to the point q — g(s) on the geodesic. We denote the lengthof the straight line to g(s) by L(s). L is clearly a continuous function

of s, with L(0) = 0.In the parametrization by arc length, the extremal g is a function

of class C2. We let 0 (O) = r<. The law of the mean gives

(3.21) g'£s) = r< + βflrftσ) 0 ^ σ ^ s

and application of equation (3.20) gives the inequality

(3.22) Ti - Ms ^ g&s) ^ r* + Ms .

Integrating (S.19) from 0 to an arbitrary point s between 0 and su

we obtain

(3.23) riS - ^ - ^ fir4(β) - flr4(0) ^ r j S2

Now suppose | r< | > Ms/2. In this case we have

(3.24) (ft(8) - ^(0))2 ^ s2(rl - Λfβ|r4| + ^ - ^ ) .

Therefore the length L(s) of the line from </(0) to g(s) satisfies theinequality

(3.25) ^ψ ^ Σ'{r\ - Ms | r, |}

the sum being taken over all r< with |r<| > Ms/2.

But

i * lrjKJfβ/2 * ~ " 4

and Σi^? = 1 implies ^ | ^ | ^ τ/7ΓHence

(3.26) 1 ^ i M . ^ 1 - M i/^Γβ - ^ ^ s 2 .s 4

Also, since the geodesic and the family of straight lines lies in apoint set N on Σ of diameter less than Δn, we have

£(s) < Δn 0 g s ^ sτ .

Now if 4n is taken to be smaller than the maximum value of thecurve L — s[l — M\/~ns — (Λf2w/4)s2]1/2 the curve L = L(s) giving thelength of the line from #(0) to g(s) must lie in a disconnected regionof the (L, s) plane. Since L is continuous and L(0) = 0, it must lieentirely in the left hand region, which shows

1044 DAVID A. POPE

(a) s < s for all geodesies in the set N.(3.27) -

(b) ί&U

Thus Lemma 3.7 follows.Now we can demonstrate that requirement 4 of 1.1 is fulfilled.

LEMMA 3.8. For any real number a, and any e > 0, there is aninteger N such that for n > N, and for all rj in Fa, we have

(3.28) F(pny) £ F{η) + e .

Proof. Let us denote the broken extremal associated with pjη bygjj. Lemmas 3.6 and 3.7 state that given any p > 0, δ > 0, there isan integer N such that if n > N, we have

f M) 0, if δ

and p are chosen sufficiently small, we will then have

(3.29) \F(pn7})-F(guη)\<e.

But from the definition of F and the remark of § 3.5 about thefundamental distance in Σa we have

(3.30) F{gnη) g F{η) .

Addition of inequalities (3.29) and (3.30) give the conclusion of the lemma.Now the homotopy described in property 5 of 1.1 will be set up.First we describe the standard deformation Θ(Ύ], U) of Morse. Let

η be any curve in Fa. Let q — q(μ) be the parametrization of η in termsof μ-length. Taking the points qo — q(t3-fa) of the approximation pn ofη as corner points, we deform η onto a broken extremal g consisting ofthe unique extremals from q, to qί+19 j = 0,1, , n.

This deformation is defined as follows: let μ3- = t^, the value of μcorresponding to the point qά. Let Δμά = μj+1 — μjm Then at time u,0 g u ^ 1/2, we take μ^u) — μ5 + 2uJμjf and construct the unique ex-tremals from μ, to μf(u). The curve g(η, u) is then defined as the curveformed by these extremals from μ5 to μj(u) and the original curve q(μ)from μά{u) to μj+1.

We now apply this deformation to the polygonal curve pjj]), defor-

ming it onto the same broken extremal g. If we set u = 1 — u in thislatter deformation, and follow the first deformation by the second, wehave a deformation θ(η, u) which carries rj to g and then to pn(if) for0 <Lu ,1. During the first half of this deformation F is not increased,and during the second half F is not decreased; thus the deformationtakes place in the set Fb, where 6 = max {F(rj), F(pn)}. But in Lemma3.8 it was shown that for n sufficiently large, b ^ a + ε for any ε > 0,η$Fa.

The deformation θ(η, u) is a homotopy, since it is easily seen to becontinuous in both η and u. Thus we have shown that the parametricproblem with the approximations described above satisfies the propertiesof §1.1.

BIBLIOGRAPHY

1. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton UniversityPress, 1952.2. M. Frechet, Sur une representation parametrique intrinsique de la Courbe continuela plus generale, Journal de Math., 9th Series, 4 (1925), 281-297.3. H. Lewy, Aspects of the Calculus of Valriations (Notes by J. W. Green), Universityof California Press, 1939.4. M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ.No. 18, 1934.5 >, A special parametrization of curves, Bull. Amer. Math. Soc. 42 (1936), 915-922.6. , Functional Topology and Abstract Variational Theory, Memorial des SciencesMathematiques, no. 92, Paris, 1939.7. ., Rank and span in functional topology, Annals of Math., 4 1 (1941), 419-454.8. T. Rado, On the problem of Plateau, Ergebnisse der Mathematik, Chelsea, 1951.9. f Length and area, Amer. Math. Soc. Colloq. Publ. No. 30, 1948.10. L. Tonelli, Fondamenti di Calcolo delle Variazioni (2 vols.), Bologna, 1925.11. L. Vietoris, Uber den hoheren Zusammenhang kompakter Raume und eine Klasse vonzusammenhangstreuen Abbildungen, Mathematische Annalen, 97 (1927), 454-472.

UNIVERSITY OF CALIFORNIA, DAVIS AND LOS ANGELES

THREE SPECTRAL THEOREMS FOR A PAIR OFSINGULAR FIRST-ORDER DIFFERENTIAL EQUATIONS

BERNARD W. ROOS AND WARD C. SANGREN*

1Φ Preliminaries* In the regular case the classical method ofobtaining eigenvalues and eigenfunctions of the equation

(1) y"(x) + [\ - q(x)]y = 0 [' = -J- ]

under Sturmian boundary conditions involves the use of asymptoticexpansions. For the singular cases of (1) when the range of x is infiniteor semi-infinite instead of finite, Titchmarsh [6] has shown that suchasymptotic solutions are also necessary in obtaining spectral and expansiontheorems by the method of complex variables. The objective of thispaper is to generalize for a particular case these types of results to thefollowing pair of equations

u'(x) - [Xa(x) + b(x)]v(x) = 0 ,

v\x) + [Xc(x) + d(x)]u(x) = 0 .

Interest in this system arises from a consideration of the Dirac relativisticwave equations for a particle in a central field. The equations (2)correspond in this case to the radial wave equations. Conte and Sangren[2] and the authors [3] have shown that most of the results of Titchmarshcan be generalized for (2) over the interval (0 g= x < °°), under therestriction a(x) — c(x) — 1. Also, the spectral properties of (2) for a(x) —c(x) = 1 over the infinite interval (— oo, oo) have been investigated [4].In this paper a discussion of the system (2) for a(x) = x2Jΰ, c(x) = x~2k

over the interval (0, oo) is presented. It is assumed throughout, thatk is a nonzero integer.

Let φ(x, λ) = [φλ{x, λ), φ2(x, λ)] and θ(x, λ) = [θ^x, λ), Θ2(x, λ)] be twosolutions of system (2) over the interval a ^ x S b, where a > 0 andb < co, such that φβ, λ) = 1, φ2(l, λ) = 0, θtf, λ) = 0, Θ2(l, λ) = 1, wherea ^ I ^b. It can be shown that the Wronskian Wx[φ, θ] = φλθ2 — φ2θ1

is independent of x so that since Wτ{φ, θ] — 1, φ(x, λ) and θ(x, λ) arelinearly independent. For the singular case it can be shown that forcomplex values of λ the system (2) has a solution ψ(x, λ) = [ψlf ψ2] =θ(x, λ) + m(X)φ(x, λ). A limit circle case is determined separately at eachof the end points, 0 and oo, by the conditions that all functions c\ψ\2

+ a\ψ2\2 are integrable, that is, belong to the class L(0,1) or L{1, oo).

In the limit point case, at either end, there exist only one ra(λ) and

Received January 6, 1961. Presented to the American Mathematical Society, PasadenaCalifornia, November 19, 1960.

* Present address: Computer Applications, Inc., San Diego, California.

1047

1048 BERNARD W. ROOS AND WARD C. SANGREN

ψ(x,X) where clψ^2 + a\ψ2\2 is integrable. The existence at each end

of at least one such integrable function is guaranteed by the directextension of WeyΓs limit-point, limit-circle theorem [6,1, 5]. Furthermore,the m(λ) at either end is such that m(λ) = m(λ) and m(λ) is analyticin either the upper or lower half plane. The spectral properties can beobtained from these functions, mo(λ) and m^X).

2. The nature of the spectrum; interval (1, oo). To obtain thespectral properties of the system (2) over the interval (1, oo) the orderproperties as x —> oo of the solution vectors φ and Θ have to be investigated.This is most easily done by using the corresponding integral equationswhich may be obtained by the method of variation of coefficients. Itis easily verified that if ^(1) = 1, φ2(l) = 0, ^(1) = 0 and Θ2{1) = 1 theappropriate integral equations are given by

(4) φx(x, X) = G^Xx) + V{

+ dφ1[H1G1(\x) - GΆiXx)]} ds

= Gλ(Xx) + L{x, φlf φ2)

(5) φ£x, X) = G2(Xx) + ^{bφ2[H2G2(Xx) - Q2H2(Xx)]

+ dφHHiG&x) -G^iXx)]} ds ,

= G2(Xx) + K(x, φl9 φ2)

θfa, X) = Hλ(Xx) + L(x, θ19 θ2)

Θ2{x, X) = H2{Xx) + K{x, θlf θ2)

where

Gx{xx) = +^ [J.ip^Δ

G2(Xx) = +*-^l [J.(p_1)(λ)Jp_1(λa?) -Δ

Hλ(Xx) = +^Δ

H2(Xx) = KlΔ

Δ =πλ

and

THREE SPECTRAL THEOREMS 1049

Let λ == σ + it where t > 0, and let

h±{x, X) = x~"l6e'"t9φ1{x9 X) and h2{x, X) — x*ertxφ2(x, λ) .

The well-known asymptotic formula Jp(z) = (2/τr^)1/2 cos (2 — 1/2 pπ — τr/4)+ 0{le~izHzl-dl2} for 0 ^ arg 2 ^ τr/2 and 2 > 1 is used in the following.The substitution of hx and h2 in (4) and (5) and the taking of absolutevalues yields

λ)!; \h2(x,X)\ ^ M + j ' t lΛllΛ,! +

where

{Λ.1/2

f- [J,(λΛ)/_p+1(λ) + JL

= max

fir2 = max {[J-p(Xx)Jp(\s) — Jp(Xx)J-p(Xs)];X—*oo

[J-p+1(Xx)Jp(Xs) + Jp^(Xx)J^.p(Xs)]}

Ml e-t{χ-s)d(s)s2k+ll2\ .

According to the lemma in [3] this can be reduced to the inequality

h^x, λ); h2(x, X) ^ Mexp j I (gx + #2) ώsi .

Consequently, it can be concluded that when b(x)x~21c and d(x)x21c are bothL(l, 00), Λi(a?, λ) and h2(x,X) will both be bounded for all values of a?.Moreover, for large x

φx(xf X) = 0{eίίCα;fc}, ^2(a;, λ) = 0{etxx~k} .

Furthermore, when Im X — 0, that is for real λ, and x —• 00 theorder properties for φ and ^ may be written as follows:

φ1(Xf x) = aj'j/iίλJJpίλa?) + v(X)J-p(Xx) + o

) ^(α?, λ) = xp{ξ(X)Jp(Xx) + η(X)J-p(Xx) + o

Θ2(x, X) -

where

(7)

λβ)] dβ]

ds]

ds]

ds] .

The integrals in (7) converge uniformly in λ hence, μ(X), v(λ), |(λ), Jj>(λ)will be continuous and bounded functions of λ.

When λ is complex and has an imaginary part greater than zero,ImX > 0, and when b(x)x~** and d(x)x2k are both L(l, «>), one can obtain

Φl%, λ) = xH-

( 8 )

θx{x, λ) =^2(a;, λ) = α!- β-<λ [iV2(λ)

where

Λf(λ) i Z 7 Γ Λ ; J β ^ / - , + i ( λ ) + iJp-xίλ) cosΔ I

+ I ^(s)^^^)^-^4-1 [i J-.p+1(Xs) + Jp-iίλs) cos kπ] ds

+ I Cί(s)^1(s)s2> [JP(XS) COS fc7Γ — /-^(λs)] d,8>

M2(λ) - ^^1—I \ + J-P+1(X) - iJv-^X) cos kπ

Δ I

+ 1 6(s)^2(s)s~2 ) + 1 [ J-p+iίλs) — iJp^(Xs) cos AJTΓ] d s

/QX — \ d(s)φ1(s)sp [ίJp(Xs) cos A TΓ + J^p(Xs)] ds>

2Vi(λ) = i ^ ^ — \ -iJ-p(X) + JP(X) cos kπΔ I

+ \°°b(s)θ2(s)s-p+1 [iJ-p+1(Xs) + Jp-As) cos

+ Γd(s)^1(s)s2) [Jp(Xs) cos &7Γ - iJ_p(λs)] ds\

iV2(λ) - ( ^ ^ β U - p ( λ ) + iJp(x) c o s & 7 Γ

+ [°°b(s)θ2(s)s-p+1 [J-p+1(Xs) — iJp^(Xs) cos kπ] ds

- [°°d(s)θ1(s)sϊ> [iJp(Xs) cos kπ + J_p(λs)] ds\ .J l J

Let ψ(x, λ) = θ(x, λ) + m(X)φ(x, λ) be that solution of (2) mentionedin the Preliminaries where c|^i | 2 + α|ψ*_2|

2 is (1, oo). ψ is given by

x(x, λ) = θx(x, λ) + m{X)φ1{x, λ) = xkeiλ* [JVΊ(λ) + m^M^X) + o(l)] ,

{x, X) = 0a(α?, λ) + m(X)φ2(x, λ) = ar f ce- ί λ* [N2(X) + m(λ)ΛΓa(λ)

It is apparent that in order for c\ψλ\2 + a\ψ2\

2 to be in L(l, oo) wemust have

do) m(χ) =M2(λ)

When I m λ - > 0 , it follows from (7) and (9)that

AΓi(λ) — (2τrσ)-1/2e«ir/2 [^(σ) + v(σ) cos A π]

M2(λ) -• (2πσ)-1/2e"ίt/2 [+//(σ) - iv(σ) cos fcπ]

Λi(λ) — (27rσ)-1/2e«*'2 [^(σ) + ^(σ) cos kπ]

iV2(λ) — (2τr(τ)-1/2eί*ir/2 [+ξ(σ) - ίη(σ) cos

Hence,

limm(λ) = _ί/ ( ) + υ(σ) cos kπ

(12) = -cos fcπ ^V-vξv\σ) + /<2(σ)

From (6) one also obtains

W.[Φ, β] = ΦA - ΦA = Avξ - μv] + o(i).

Because W0[ψ, θ] = 1,

vξ - μv = -j

The substitution of this result in (12) yields

l i m J m m ( λ ) = — σπ

*-o 7 v\σ) + μ\σ)

Hence, Im m(X) is a nonpositive, nonvanishing continuous and boundedfunction of X for both positive and negative λ.

This is easily seen for positive λ since λ, v2 and μ2 are all positivenumbers. For negative λ not only is λ negative, but so are v2 and μ2.This can be verified by making use of the formula

J9(i*»z) = ί2mpjp(z) .

The spectrum is, therefore, continuous over the infinite interval— oo < λ < oo. These results may be summarized in the following theorem,

THEOREM 1. Consider the system (2) where a(x) = x2k and c(x) = x~2k

over the semi-infinite interval [I, oo] and under the boundary conditions

(13) u(l) cos a + v(l) sin a = 0

Lei &(#) αraϊ c£(x) 6β real-valued continuous functions of x and let b(x) x~2k

and d{x)x2k belong to the class L(l, oo). A solution of the system (2),(13) is defined as a vector function [u(x, λ), v(x, X)] with continuousfirst derivatives satisfying this system. The values of X for whichsuch solutions exists form a continuous spectrum over the real X-axis(—00 < X < oo).

In order to simplify the appearance of the equations in the preceedingproof and in the following, it was assumed, without loss of generality,that a = π/2 and I = 1.

3 Nature of the spectrum; interval (0,1).

THEOREM 2. Consider the system (2) where a(x) = x2k, c(x) = x~2k overthe interval 0 ^ x ^ 1 and subject to a linear homogeneous boundarycondition at x = 1. The spectrum is discrete provided:

b(x)

b(x)

and d(x) are L(0,1)

and d(x) are L(0,1)

and d(x)x2p are L(0,1)

forp = k + 1/2

for 0 < p < 1

for p < 0 .

The proof of this theorem follows closely that of Theorem 1. Exceptfor an obvious change of integration limits, the integral equationscorresponding to system (2) of Theorem 2, are given by equations (4)and (5).

First, consider the case p > 0. For x —• 0 one has the well-knownasymptotic relations:

Γ(p + 1)2* πand for p - 1 > 0:

wLet hx(x, X) = φx(x, λ), h2(x, X) =

t > 0. One obtains:X) and X = σ + it where

+ J-p

[JP{Xs)J-p{Xx)J-p(Xx)J^{Xs)]\ ds\


λ) =

[Jp(Xs)J.p+1(Xx) + J^Xs^^Xx)]} ds\ .

For x —* 0 tφc, X) and h2(x, X) are bounded for all x provided bx~Up~1] andd are L(0,1) (see lemma [3]). Hence, φλ{xf X) = o(l) and φ2(x9 X) =o(χ-2{p-1])for ίc-^0.

It follows that one may write

φx(x, λ) = [MX{X)

φ2(x, X) = a r 1 ^

where

Mλ(X) = ( " 1 ) ) | > (ΓJ L

and

Γ - Jp-xίλ) + Γ[J3,_1(λs)δ^2s- ί>+1 - dφ1Jp(Xs)sp] ds\ .

Similarly,

θx(x9 X) - [Λi(

^2(x, λ) = »-«'-" [N2{X)

where

and

— JPM + \ {δ^2s~p+1Jr

3,_1(λs) — dθ^JJXs)} iL Jo

A similar argument as in §3, yields

Hence

Λ(λ) ~ [{bθ.s-^J^iXs) + dθ^J^Xs)} dsm(λ) = \

\1 dsp-^λ) - \1{bθ2β-*+1Jp-.1(\s) + dφ1spJp(Xs)}

Jo

For t —>0, i.e., ImX—>0, Mi(λ) and JVi(λ) tend to real limits. Thisis apparent for λ > 0 and is easily shown for λ < 0 by using again therelation Jr(zeimπ) = eίm*r J r 0 ) . Hence, Im m(λ) —> 0 except possibly atthe zeros of MΊ(λ). Consequently, the spectrum will be discrete.

For values of the parameter p such that 0 < p < 1 one has

7Γ

It is not difficult to verify, applying the same method as discussed above,that in this case, when b(x) and d(x) are 1/(0,1) the limit circle caseprevails and the spectrum is discrete, fc, an integer, is not in this interval.

For p < 0 the asymptotic expressions become

(λaQ-"-»

In a similar fashion, for the case where p > 1 one can obtain that0X(#, λ) = o(x2p) and 2( > λ) = o(l). In this situation the additionalcondition that b and d(x)x2ί) are in L(0,1) must be imposed. The conclusionthat the spectrum will be discrete follows accordingly.

4 Nature of the spectrum; interval (0, oo).In this case singularities exist at both ends of the interval and it

has been shown [4] that the spectral properties of the system (2) aredetermined by the imaginary parts of the following functions:

1 m 0 m 0 rrtoo

m 0 — moo m 0 — m ^ m 0 — m ^

where mo(λ) and m^{X) are the previously determined m's at x — 0 andoo. As was shown in the previous sections mo(λ) is a meromorphic functionwhile moo has a nonvanishing imaginary part. It is clear that theimaginary parts of the three functions above tend to finite limits whichcan vanish at most at discrete points. The associated spectrum willtherefore be continuous over the whole real λ-axis.

THEOREM 3. Consider the system (2) where a(x) = x2k and c(x) = x~2k

over the interval 0 ^ x oo. The spectrum will be continuous over theentire real X-axis (—00 < λ < 00), provided b(x) and d(x) satisfy theconditions given in Theorem 1 and Theorem 2.

REFERENCES

1. E. A. Coddington and N. Levinson, Theory of ordinary differential equations, MacGraw-Hill New York, 1955.2. S. D. Conte, and W. C. Sangren, An Expansion theorem for a pair of first orderequations, Can. J. Math., 6 (1954), 554-560.3. B. W. Roos and W. C. Sangren, Spectra for a pair of singular first order differentialequations, Proc. Amer. Math. Soc, 12 (1961), 468-476.4. , General Atomic Report (unpublished).5. , General Atomic Report GA-1777.6. E. C. Titchmarsh, Eigenfunction expansions associated with second-order differentialequations, Oxford, 1946.

GENERAL DYNAMICS CORP.

JOHN JAY HOPKINS LABORATORY FOR PURE AND APPLIED SCIENCE

GENERAL ATOMIC DIVISION

GENERAL DYNAMICS CORPORATION

SIMPLE MALCEV ALGEBRAS OVER FIELDSOF CHARACTERISTIC ZERO

ARTHUR A. SAGLE

l Introduction* Malcev algebras are a natural generalization ofLie algebras suggested by introducing the commutator of two elementsas a new multiplicative operation in an alternative algebra [3]. Thedefining identities obtained in this way for a Malcev algebra A are

(1.1) xy = ~yx

(1.2) xy xz = (xy z)x + (yz x)x + (zx x)y

for all x,y, z e A. Since Albert [1] has shown that every simple alter-native ring which contains an idempotent not its unity quantity is eitherassociative or the split Cayley-Dickson algebra C, it is natural to seeif a simple Malcev algebra can be obtained from C. In [3] a sevendimensional simple non-Lie Malcev algebra A* is obtained from C andis discussed in detail. In this paper we shall prove the following

THEOREM. Let A be a finite dimensional simple non-Lie Malcevalgebra over an algebraically closed field of characteristic zero. Fur-thermore assume A contains an element u such that the right multi-plication by u, Ru, is not a nilpotent linear transformation. ThenA is isomorphic to A*.

The necessary identities and notation from [3] for any algebra Aare repeated here for convenience:

(x, y) — [x, y\—xy — yx

(x, y, z) = xy z — x yz

J(x, y, z) = xy z + yz x + zx y

for x,y,ze A. If h(xlf , xn) is a function of n indeterminates suchthat for any n subsets B{ of A and b{ e B{, the elements hφlf •••, bn)are in A, then h(Blf , Bn) will denote the linear subspace of A spannedby all of the elements h(blf •••,&„).

For a Malcev algebra A of characteristic not 2 or 3, we shall usethe following identities and theorems from [3]:

(1.6) J{x, y, xz) = J(x, y, z)x

(1.3)

(1.4)

(1.5)

Commutator,

Associator,

Jacobian,

Received September 2, 1961. The author would like to thank Professor L. J. Paigefor his assistance in the preparation of the manuscript. This research was sponsored inpart by the National Science Foundation under NSF Grant G-9504.

1057

1058 ARTHUR A. SAGLE

(1.7) J(x, y, wz) + J(w, y, xz) = J(x, y, z)w + J{w, y, z)x

(1.8) 2wJ(x, y, z) = J(w, x, yz) + J(w, y, zx) + J(w, z, xy)

(1.9) J(wx, y, z) = wJ(x, y, z) + J(w, y, z)x — 2J(yz, w, x)

(1.10) xy zw = $(te^ 2) + w(ys #) + y(sa? w) + 2(#w y)

for all w, x, y, z e A. If N = {x e A: J(x, A, A) = 0}, then it is shownin [3] that N is an ideal of A which is a Lie subalgebra and further-more for α, b e A

(1.11) J(a, 6, A) = 0 implies ab e N.

It is also shown in [3] that J(A, 4, L) is an ideal of A. Thus if A isa simple non-Lie Malcev algebra we have

(1.12) N = 0 and A = J(A, A, A) .

We shall assume throughout this paper that A is a finite dimen-sional simple non-Lie Malcev algebra over an algebraically closed fieldF of characteristic not 2 or 3 containing an element u such that Ru isnot a nilpotent linear tansformation. In § 2 the basic multiplicativeidentities are derived using methods analogous to those of Lie algebras.Decomposing A = AQ 0 AΛ 0 0 Ay into weight spaces relative toRu [2; page 132] we prove the block multiplication identities AΛAβ cAa+β if a Φ β, Ala A-a, and Al — 0. Further identities are derived in§ 3 which lead to the important result that there exists a nonzeroweight a such that A = 4 o 0 A α 0 A-Λ where AQ— A*A-.».

In § 4 we show that R(A0), the set of right multiplications RXQ byelements x0 e A09 is a set of commuting linear transformations on thesubspaces Ao, Aω and A-Λ. Analogous to Lie algebras we decomposeA = A0Q) AωQ)A-Λ into weight spaces relative to R(A0) [2; page 133]and thus find a basis of A which simultaneously triangulates the matricesof R(A0). We now introduce the trace form, (x, y) — trace RxRy, in § 5and assume for the remainder of the paper that the algebraically closedfield is of characteristic zero. With this and the results of § 4 weeasily show that (x, y) is a nondegenerate invariant form on A =Λ Θ 4 Θ A - « and A, = uF.

In § 6 we show that Ru has a diagonal matrix of the form

Ό 0"

al

0 -al_

Using this and a few more identities we show in § 7 that the simpleMalcev algebra A = Ao 0 Aa 0 A-^ is isomorphic to the seven dimen-

SIMPLE MALCEV ALGEBRAS OVER FIELDS OF CHARACTERISTIC ZERO 1059

sional algebra A*.

2. Basic multiplication identities* Let Ru (u e A) be a fixed non-nilpotent linear transformation and decompose the simple Malcev algebraA into the weight space direct sum A = 4 o © 4 Λ 0 0 i v relative toRu where the weight space of Ru,

Aa = {x e A: x(al — Ruf = 0 for some integer k > 0} ,

is a nonzero ^-invariant subspace of A corresponding to the weight aof Ru. Let xω e Aω, xβ e Aβf then using (1.6)

J(u, xΛ, xβ)Ru = J(u, xΛ, xβ)u = J(u, xΛf uxβ) = —J(u, xa, xβRu)

and therefore

J(u, xΛ, xβ)(βl + Ru) = J{u, xΛ, xβ(βl - Ru)) .

Now letting yβ = xβ(βl — Ru) e Aβ we have

J(u, x«y xβ(βl - Ruf) = J(u, xΛ, yβ{βl - Ru))

= J(u, xω, yβ)(βl + Ru)

= J(u, xai xβ(βl - Ru))(βl + Ru)

= (u, xa, xβ)(βl + RJ .

Continuing by induction we obtain

(2.1) J(u, x«, xβ)(βl + RUY = J(u, xΛ, xβ(βl - Ru)n)

for every integer n. Since xβ e Aβ there exists an integer N suchthat 0 = J{u, xaf xβ(βl— RU)N) = J(u, xa, xβ)(βl +RU)N and this showsJ(u, xΛ, xβ) e A_β. Now interchanging the roles of xβ and xa in (2.1)we also obtain J(u, xm xβ) e A-a and thus

(2.2) J(u, AΛ, Aβ) c A-Λ Π A-β .

From (2.2) we have the following relations

(2.3) J(u, Am AΛ) c A_Λ

(2.4) J(u, Am Aβ) = 0 if a Φ β .

We shall now prove

(2.5) AaAβ c Aa+β if a Φ β .

For if a Φ β and xΛ e Am xβ e Aβ we have by (2.4),

0 = J(n, xa, xβ) = (xΰύxβ)Ru — xjtu xβ — x« xβRu

that is, (xaxβ)Ru = xaRu xβ + x* %βRu and so Ru is a derivation of

AcbAβ into AaAβ. This yields

{xaxβ){Ru - (a + β)I) = &Λ(iJtt - α l ) α

and in the usual was we prove the Lebnitz rule for derivations whichthen yields that for some integer N, (x«xβ)(Ru — (a + β)I)N = 0 andtherefore xobxβ e Aω+β. In particular we have

(2.6) A0A«c:Aa if a Φ 0 .

We shall now investigate Ao more closely. Let xa e Aa9 xβ e Aβ and#0 e Ao, then by (1.7) J(xOf xβ, uxω) + J(w, xβ, xQxa) = J(x0, xβ9 xa)u +J(u, xβf xa)xQ. Therefore if 0 Φ a Φ β we have by (2.4) J(xQ, xβ, uxa) =J(x0, xβ, xΛ)u. This yields J(x0, xβ, xjfiίl — Ru)) = J(x0, xβ, xΛ)(aI + Ru)and as in the proof of (2.4) we obtain

(2.7) J(A0, Aa, Aβ) = 0 if 0 ^ α Φ β Φ 0 .

Next let x09 y0 e Ao and xΛ e AΛ where a Φ 0, then using (1.9),(2.4) and (2.6) we have

J(xQu, yQ, xΛ) = x0J(u, y0> xΛ) + J(xQ9 y0, xa)u - 2J(yQx«, x09 u)

and in general we have J(x0R%, y0, xa) = J(#o, l/o> β») Bί which im-plies J(a?0, y0, »„) G Ao. Now by (1.7), J(^ o , y0, uxΛ) + J(u, yQ, xoxa) =J(a?0, i/o,««)% + J(u9 y0, x*)x0; and using (2.4) and (2.6) we obtainJ(x0, y0, xaRu) = —J(x0, Vo, x«)Ru which implies J(x0, y0, xa(Ru - al)) =—J(xOf y0, x<x)(Ru + oil). Thus, as usual, we have J(x0, y0, xa) e A-a andtherefore J(x0, y0, xΛ) e Ao Π A_Λ which proves

(2.8) J ( Λ , Λ , Λ.) = 0 if a Φ 0 .

We shall now show Al c Ao. From our basic decomposition A =Λ Θ Λ Θ Θ Λ relative to Ru we can find a basis {xx{τ)y , w(τ)}(t^ = mτ) of AΓ such that

(2.9) Xι(τ)Ru = Σ aij%Aτ) + ^ ( τ )

where τ, α^ e F and i = 1, , m. In particular let {^(0), , a{xlf •••,.»«} be the above type for AQ. Then xjiu = 0 and

χ.jζw _ γ^aikxk (i = 2,

Furthermore,

£, ajt , a?y) =

with the understanding that α10 = 0.Using (1.6) and operating on both sides of the previous equation

with Rl, we obtain

i - i

Now by assuming i < j and choosing n large enough, a simple induc-tive argument yields %&5 e Ao for all i and j . Thus Ao c Ao.

Using (1.8), A0

2 c Ao and (2.8) we have

AΛJ(AQ, Ao, Ao) c J ^ c , Ao, Ajj) c /(A*, Ao, Ao) = 0 for ^ ^ 0 .

i n u s , ^±«y (^i0, u4.0, -Ao) c ^JQJ A^J (AO, AOf Ao) — A0J \A0, J±09 Ao) cz «y(A0, A.Q, A.Q),

or J(^40, AQ, AO) is an ideal of A. But since /( io, AQ, Ao) c AQΦ A andA is simple we have

(2.10) J(A0, Ao, Λ) = 0 .

Now using (2.8) and (2.10) we have J(A0, Ao, A) = Σ » (Λ> Λ , ^«) =0 and by (1.11) and (1.12),

(2.11) AlcN=0 .

In particular this means the kernel of Ru is 40.We shall now show A\ c A_Λ. Let xΛ, ya e Aa for a Φ 0, then by

(2.3) / ( ^ , a?rt, 2/J = (xaya)Ru + 2/<Λ xa + yΛ α;ΛieM = w^ e A-a . There-

fore (xaya)Ru — %«RU y<* + y«- y<*Ru + w-<* which yields

(x»V.)(Ru - 2αl) - &„(!?* - α Z ) . yΛ + xa . yΛ(Λβ - α l ) + wL1^ .

By induction we obtain

where ί_%2 6 A_*. Therefore for large enough ΛΓ, (x^y^iRu — 2^/)^ 6 A_Λ.Now let ajΛj/Λ = Σ Y V where 2:γ e Ay, then (α?Qί7/0>)(JBw — 2aI)N =Σ Y «γ(βf» — 2α/)* G A-β,. Therefore by the ^-invariance of the Ay andthe uniqueness of the decomposition A = AQ φ Aa 0 0 Aλf ^γ(i2tt —2a/)^ = 0 if y Φ —a. Thus if y Φ —a,zy e A2». Therefore α;^^ =z2a + ^_Λ which proves


LEMMA 2.13. J(u, A%, A2a) = 0.

Proof. Using (2.12), (2.7) and (2.3) we have

J(u, Al, A2cύ) c J(u, A-m A2cύ) + J(u, A2Λ, A2cύ) c J(u, A2cύ, A2cύ) c A_2QJ .

Now for any x, y e Aa, z e A2oi we have by (1.7) J(z, u, xy) + J(x, u, zy) =J(z, u, y)x + J(%, u, y)z and using (2.4), (2.5) and (2.3) this yieldsJ(z, u, xy) = J(x, u, y)z e A-a A2ΰύ c Aa. Combining these results wehave J(u, Al, A2a) c Aa Π A_2α> = 0.

Now let w 6 A2m x,y e Aa and xy — z20b + 2_Λ where z2ΰύ e A2ΰύ,Z-a 6 A-a, then using Lemma 2.13 and the fact J(u, A-m A2cύ) = 0 wehave

0 = J(u, xy, w) = J(u, z2a, w) + J(u, £_„, w) = J(w, 2;2α, w)

that is,

J(u, z2m A2Λ) - 0 .

Now since z2oύ e 42OJ we also have by (2.4) J(u, z2oύ1 Aβ) = 0 if /3 ψ 2a.Combining these results, J(u, z2m A) = Σβ J(u$ z™, Aβ) = 0 and there-fore z2au e N= 0 by (1.11) and (1.12). Thus 0 = z2aRu and thereforez2cύ e 4 0 ί l A2α> = 0 and this proves

(2.14) Al c A_* .

Also note that we now have

(2.15)

3, More identities• Let A = Ao 0 Aα 0 0 Ay be the decomposi-tion of A into a weight space direct sum relative to Ru and supposethat for weights α, /S, 7 of 22W, β Φ j and β + j Φ a. Then for a? e AΛ,y e Aβ and ^ e Aγ we have by (1.9) and (2.4)

J(xu, y, z) = α;J(%, 1/, «) + J(x9 y, z)u — 2J(yz9 xy u) = J{x, y, z)u

and therefore J(x(Ru — al), y, z) = J(x, y, z)(Ru — al). By inductionwe have J(x(Ru — al)n, y, z) — J(x, y, z)(Ru — al)n and hence

(3.1) J(AΛ, Aβ, Ay) c i α if β Φ 7 and β + Ί Φ a .

By the symmetry of the a, β and 7 we may also conclude

(3.2) J(Aβ, Ay, Aa)(zAβ if 7 Φ a and 7 + a Φ β

(3.3) J(Ay, AΛ9 Aβ)dAy if a Φ β and a + β Φ 7 .

Now assume a Φ β Φ y Φ a. Suppose /3 + 7 — a. If 7 + a = /3,

then 7 = 0 and therefore a = β, a contradiction. Therefore 7 + cc φ βand by (3.2) J(Aβ, Ay, Aa) c Aβ. Similarly if a + β = 7, then β = 0and α = 7, a contradiction. Therefore a + β Φ 7 and by (3.3)J(Aγ, AΛ, Aβ) c Ay. Thus we have J{Aa, Aβf Ay) c Ay Γi Aβ = 0 if a Φβ Φ 7 Φ a and /5 + 7 = <*.

With the assumption a Φ β Φ 7 Φ a, suppose now that β + j Φ a.Then by (3.1), J{Aa, Aβ, Av) c Aα. We next note that it is impossibleto have 7 + oc = β and a + β = 7. So using (3.2) or (3.3) togetherwith J(Aωf Aβ, Ay) c Aω we conclude /(A*, Aβ, Ay) = 0. Thus we canconclude, using the preceding paragraph,

(3.4) J{AΛ, Aβ, Ay) = 0 if α =£ /3 7 ^ a .

Now assume two weights are equal, that is, a = β. Suppose 7 Φ0, a, —a or 2α, then

U yjΓXtfj -Li-cύf L*-y) — ** cύ ^*-y 1 L*-oύ-L*-y * " - α > " 1 "

c A_α

However using (3.1) J{Aa, AΛf Ay) c Aa and therefore J(Aα, A^, Ay) c

Aα Π (A_α+Y 0 Ay+2ΰύ) = 0. This proves

(3.5) J{AΛ, Aa, Ay) = 0 if 7 ^ 0, α, or - α 2α .

For the "exceptional" cases we have

(3.6) J(A«, Aa, Aa) c A\ AΛ c A-βΛ* c Ao .

(3.7) J(Aα, AΛ, Ao) c A1Λ + AaAQ. A. c A_α .

(3.8) /(A,,, AΛ, A_*) c AM-« + A^A.. Aa c AΛ .

(3.9) J(AΛ, Aα, A2α) - 0 .

To prove (3.9) let x,y e Aa,ze A2α>, then by (1.9), (2.5) and (2.4)

J(xu, y, z) = χj(u, y, z) + J(x, y, z)u — 2J(yz, x, u)= J(%, y, z)u

and as usual we have J(x(Ru — al)n, y, z) = J{x, y, z)(Ru — al)n. There-fore J(x, y, z) e AΛ. However by (1.7) J(x, y, uz) + J(u, y, xz) =J(x, y, z)u + J(u, y, z)x and using (2.4) we obtain J(x, y, uz) — J(x, y, z)u.This yields J(x, y, z(2al — Ru)

n) = J(x, y, z)(2al + Ruf and thereforeJ(x, y, z) e A_2Q>. Combining the above results we have J{x, y, z) e Aa ΠA_2α> = 0 if a Φ 0.

We shall now show A^Aβ — 0 if a Φ 0 and β Φ 0, ±a. Let α and/5 be fixed weights of Ru and assume β Φ ka, k = 0, ± 1 , ± 2 , , with


a Φ 0. Then for any other weight γ we have by (3.4) J(Aβ, Aay Ay) =0 if β φ a Φ γ Φ β. However a Φ β and therefore J(Aβ, Aa, Ay) =if a Φ γ Φ β. Suppose γ = α, then by (3.5) and the choice of β,J(Aβ, Aa, Aa) = 0. Suppose γ = β, then J(Aβ, Aβ, Aβ) = J{Aβj Ae, Aa) =0 if α =£ o, β, -β or 2/9. We know a Φ 0, β or -/3 so if α: = 2/5, thenby (3.9) J(Aβ, Aβ, Aα) = 0. Combining all these cases we have shownJ(Aβ, AΛ9 Ay) = 0 for any weight γ and therefore J(Aβ, AΛ, A) =Σ Y J(Aβ, A«, Ay) = 0. By (1.11) and (1.12) AaAβ c N = 0. This proves

(3.10) AaAβ = 0 if a Φ 0 and β Φ ka, k = 0, ± 1 , ± 2 , .

We now assume a Φ 0 and β = ka for & =£ 0, ± 1 , then J04*, Aβ, A7) =J(Aa, Akoύ, Ay) = 0 if a Φ ka Φ j Φ a, by (3.4). But since k Φ 1 wehave J(Aα, A^ , Av) = 0 if a Φ 7 Φ ka. Suppose γ = a, then using (3.5)

Aβ, Ay) = J ( A Λ , i4 A Λ , Ay)

- 0

if &α: 0, α:, — a or 2#. But by the choice of k we need only considerka = 2α and in this case J{AΛ, AΛ, Akoύ) — 0 by (3.9). Now supposey = ka, then

#, Aβ, Ay) = j ( A α , A^^, Aγ)

Akcύ, Akoύ)

- 0

if α: Φ 0, &α, — fcα: or 2ka, by (3.5). Again by the choice of k and awe need only consider a — 2ka. In this case k — 1/2 and thereforeγ = £ = ka = l/2a. This yields J{Am Aβ, Aγ) = J(Aβ, Aβ, A23) - 0 by(3.9). Combining all of these cases we have for any weight γ,J(Aa, Akcύ, Ay) — 0 if a Φ 0, k Φ 0, ± 1 and as before this gives

(3.11) A Λ A , . - 0 if aΦθ, k Φ 0, ± 1 .

(3.10) and (3.11) yield

(3.12) AaAβ = 0 if a Φ 0, β Φ 0, ± α .

Since iϋω is not nilpotent, there exists a weight a Φ 0. We shallnow show that —α is also a weight of iϋw. For suppose — a is not aweight, then by the usual convention A_Λ = 0 and noting that none ofthe previously derived identities use the fact that A_Λ Φ 0 we have forβ Φ 0 or a, that A«Aβ = 0 by (3.12). For £ = 0, AΛAβ c A0 and for


β — a, AoύAβ c A - β = 0 using (2.14). Therefore Aa is a nonzero ideal ofA and so A = AΛ. But u e A and ujϊ Aa — A, a contradiction. There-fore —a is a weight if a is a weight.

Now set J ^ = AΛA_a φ i t f φ A-* where a is a nonzero weight.Then s/& Φ 0 and for /3 = 0, ±a we have J*£Aβ c s*ζ. For /5 =£ o, ±awe have AαAβ = A-aAβ = 0 by (3.12). Now by (3.4) and (3.12) we havefor x e AΛf y e A_Λ, 2 € Aβ that 0 = J(x, y, z) = xy > z + yz x + zx * y ~xy £ and so 0 = AaA-a Aβ. Thus in all cases S$ζAβ c J^J and there-fore s/co is a nonzero ideal of A and we have A = j^£. This proves

PROPOSITION 3.13. If A is a finite dimensional simple non-Lie Malcevalgebra over an algebraically closed field of characteristic not 2 or 3and A contains an element u such that Ru is not a nilpotent lineartransformation, then there exists an a Φ 0 such that A — Ao 0 AΛ 0 A-a

where Aa — {x e A: x(al — Ru)k = 0 for some A; > 0} and Ao = AaA^Λ.

4 A decomposition of 4 relative to ^40. Let us consider the de-composition of A as given Proposition 3.13; that is,

-ti- — -tio M7 - -α vi/ ^±_φ

For any j / 0 , z0 e Ao and a; e Aa(a = 0, ± α ) , we use (2.8) and (2.11) tosee that

0 = J(x, Vo, z0) = α;(i?^o - Rzβy) .

Therefore,

is a commuting set of linear transformations acting on Aa. We canfind i2(A0)-invariant subspaces Jlfλ(α) [2; Chapter 4] such that

Aα = Σ θ ΛΓχ(α) (α - 0, ± α ) ,λ

where on each Mλ(a) the transformation RXQ, for any α;0 6 Ao, has amatrix of the form

( 0 ) oL *

that is, ikfλ(α) has a basis {a?lf a?2, , xm} (m = m(λ, α)) such that forany x0 e AQ, there exists α o (α?0) e F for which

(4.1) ajiie = Σ αii(»o)«i + λ ί ^ ) ^ ,

where λ(a?0) 6 F and, of course, i = 1, 2, , m.


Using the usual terminology we call the function λ defined byλ: x0—> X(x0) a weight of Ao in Aa or just a weight and the correspondingMλ(a) a weight space of Aa corresponding to λ or just a weight spaceof Aa. It is easily seen [2] that Aa has finitely many weights and theweights are linear functionals on Ao to F. Also

Mλ(a) = {xe Aa: for all x0 e A , x(RXQ - \(xo)I)k = 0

for some integer k > 0}

and for this weight λ we have X(u) = a. For suppose X(u) = b, thenthere exists an x Φ 0 in Mλ(a) such that bx — xRu. But Mλ(a) a Aa —{x £ A: x(Ru — al)n = 0}; therefore (6 — a)x = a?(l?tt — α/) and by induc-tion (δ — a)nx — x(Ru — aiy so for some integer JV, (b — a)Nx =x(Ru — α/)^ — 0 and thus a = δ = λ(^). We now combine the weightspace decompositions of the Aa to form a weight space decompositionof A in

PROPOSITION 4.2. Let 4 = A o 0 i « 0 A-a be a simple Malcev alge-bra as determined by Proposition 3.13, then we can write A = Ao ®Σλ®-Mλ(^)® Σj*Θ-Mμ(~"α) where all weights are distinct and anynonzero weight p of Ao in A is a weight of AQ in Aa or A-a but notboth.

Proof. The first part is clear noting that in the original weightspace decomposition Aa — Σγ Θ Λfy(α) the weights of Ao in Aα can betaken to be distinct. Also if λ is a weight of Ao in AΛ and μ a weightof Ao in A_α, then λ(^) = a Φ —a = μ(n) and therefore λ Φ μ. Nowlet |O Φ 0 be any weight of AQ in A with weight space Mp ={xe A: x(RXQ - ρ(xo)I)k = 0} and let y = yo + ya + y-»e Mp where ya e Aa

with a = 0, ±a. Then for some integer N> 0,

0 =

and by the uniqueness of the decomposition A — AQ 0 Aω φ A_Λ wehave 2/β(i2β0 — p(xo)I)N = 0 for α = 0, ± α . Now by using the binomialtheorem and A2

0 — 0 we have 0 = yo(RXQ — p(xo)I)* — yop(xo)N and since

P Φ 0, yQ = 0. Thus we have ί/α(i2ίCo — p(xQ)I)N = 0, α — ± α , for someinteger JV and so /> is a weight of Ao in AΛ and A_α. Now suppose yΛ

and 2/_Λ are both nonzero, then since p is a weight of -Ao in AΛ, ρ{u) —a and since p is a weight of Ao in A_α, p(u) = — a, a contradiction.Thus /> is a weight of Ao in either AΛ or A-a but not both.

We shall use the usual convention that if p is not a weight of Ao

in A, then ikΓp = 0. Let Mx{a) and Mμ(α) be weight spaces of Ao in Aa


and let x0, y0 e Ao and x e Mλ(a), y e Mâ), then using (2.8) and (1.7)we have

J(x, x0, yoy) = J(y0, x0, xy) + J(x, x0, yoy)

Thus J(x0, x, y(Ry0 - μ(yo)I)) = -J(%0, %, y)(RyQ + iKvÎ) and by induction

J(x0, x, y(Ryo - μ(yo)iy) = (- l )V(s 0 ,

From this we obtain J(ô, x, y) e Af_μ(—α) and interchanging the rolesof a? and y we see J(a?0, x, y) e M-λ(~a); this proves

(4.3) J ( Λ , Mλ(α), Mμ(α)) c ikf_λ(-α) ΓΊ Λf_μ(-α) .

From (4.3) we obtain

(4.4) J(A0, Mλ(α), Mλ(α)) c ikf_λ(-α)

(4.5) J ( Λ , ilfλ(α), Λfμ(α)) = 0 iί X Φ μ .

We shall next show

(4.6) Mλ(a)Mμ(a) = 0 i ί X Φ μ .

For let x0 e Ao, α? e Mλ(a) and y e Mμ(α), then by (4.5) 0 = J{x, y, x0) andtherefore xyRXQ = î2χ0 y + a? 2/i2χ0 and hence xy(RXQ — (μ(x0) + λ(a?0))/) =^(i2Xo — λ(a?0)/) •!/ + «• 2/(-Bχ0 — μ(xo)I). In the usual way we can provethere exists an integer N such that xy(RXQ — (μ(x0) + X(xo))I)N = 0 andsince we know xy e A-a this shows xy e Mλ+[l(—a) if λ + μ (defined by(λ + μ)(x0) = λ(#0) + μ{x0)) is a weight of Ao in A_α, or a?2/ = 0. Ifxy φ o, then λ + μ is a weight of Ao in A_α where λ and μ are weightsof Ao in Aa and therefore — a = (λ + /£)(u) = λ(u) + /ί(u) = a + α, acontradiction.

Next we have for any weight λ of Ao in Aa

(4.7) Mλ(α)Mλ(α) c ikf_λ(-α)

if — X is a weight of Ao in A_α. For let xQ e Ao and λ == X(x0) e F andlet Mλ(a) have basis {xlf •••, ccm} as in (4.1). Then using (1.2) we obtain

and thus

0 = x^2(i2,2

n - XRXn - 2XU) = ^ ,(22^ + λl)(2?.o - 2λl) .


Now since λ is a weight of Ao in Aa, —2λ is not a weight of Ao inA-a: —a = (2λ)(^) = 2X(u) = 2a. Thus the above equation impliesx1x2(RXQ + λJ) = 0 and therefore xλx2 e MLλ(—a). Next xtx0 x3x0 =Xx^XXz + a32x2 + aslxτ) — X2xxx% + s where s e M_ λ(—a) and (xo#i Xz)%o +(x1xs x0)x0 + (α?3Xo ô)î = — λΈi#3-R*0 + 1 3-B π + λâsx + f where ί e ikf_λ(—α).Therefore using (1.2) we obtain 0 = x&^R^ + Xl)(RXQ — 2X1) + w wherew e M-λ(—a) and actually w = SXαgâ?!. Therefore 0 = a?1a;3(Jίa.n + λ J ) 2

(i?Xo — 2λ7) and as before x&^R^ + λ/) 2 = 0 so t h a t xxxz e M_ λ (—α).Continuing this process we obtain xλxk e ML λ(—α) for fc = 1, 2, •••, m.Next consider the product

XzX0 = (λa?2 + αala?i)(λfl?3 + α 3 2x 2

= X2X2X3 + S

where s e ikf_λ(—α) and

(α?0^2 ^ 3 ) ^ 0 + (^2α;3 xo)xQ + (x,x0 α;0)^2 = x2x^(R!0 - \RXQ - λ2/) + t

where t e M_λ(—α), therefore 0 = x*%JίRXQ + Xl)(RXQ — 2λ/) + w where11; 6 ikf_λ(—α). Therefore for some integer & > 0 such that w(RXQ + λ/)fc —0 we have 0 = x2x3(RXQ + Xl)k+1(RXo — 2X1) and as before x2x3 e M l λ ( — a ) .We continue this process showing x2xk e M-k(—a) and in generalXiXj e M-λ(—a) for i, j = 1, •••, m. This completes the proof of (4.7).

We now show

(4.8) Mλ(a) Mμ{-a) = 0 if λ + μ =£ 0 .

By (2.7) we have for x e Mλ(a), y e Mμ(—a) and #0 e Ao that 0 —J(x, y, x0) and as usual we obtain xy(RX(i — (λ(α?0) + μ(xo))I)N = 0 forsome integer N> 0. Now 2 = #2/ e Ao and suppose 2 =£ 0, then, sinceX + μ Φ 0, X + μ is a nonzero weight of Ao in ^40, a contradiction toProposition 4.2.

Let x e ikfp(α), 1/ e Mλ(a) and 2 e Λfμ(—α), then using (1.9), (2.7) and(2.8) we have

/(αô, y, z) = a?J(a?o, 1/, 2) + «/(&, 1/, ) o - 2J(yzf x, x0)

= J(χ, y, z)x0

and therefore J(x(RXQ — p(xo)I), y, z) = J(x, y, z)(RXQ — ρ(xo)I) and asusual we obtain J(x, y, z) e Mp(a). Interchanging x and y we also obtainJ(x, y, z) e Mλ(a) and therefore J(x, y, z) e Mλ{a) n Mp(a) = 0 if λ Φ p.Now assume λ Φ p and assume μ = — λ is a weight of Ao in A_α, then

0 = J(cc, y,z) = xy * z + yz x + zx y — yz * x ,

using (4.6) and (4.8). This proves


(4.9) Mλ(α)M_λ(-α) . Mp(a) = 0

if λ Φ p are weights of AQ in Aa such that — λ is a weight of Ao in

We shall now show if λ is a nonzero weight of Ao in Aa withweight space Mλ(a), then — λ is a nonzero weight of Ao in A_α withweight space M_λ( — a). The proof is similar to that following (3.12):Suppose — λ is not a weight of Ao in A_β, then M_λ(—a) = 0; Mx(a)Mk(a) =0; Mλ(α)ikfp(α) = 0 if iO Ψ λ; ΛMλ(α) c Λfλ(α) and Mλ(a)M^(-a) = 0 ifμ + λ ^ 0. Thus Λfλ(α) is a proper ideal of A, a contradiction.

Set ikfλ = Mκ(a)M-λ(—a)Q) Mλ(a) 0 Λf_λ( —α) for some nonzero weightλ of Ao in AΛ. Then analogous to Proposition 3.13, Mλ can be shownto be a nonzero ideal of A and we have

PROPOSITION 4.10. If A = A 0 A*0^-« is a simple Malcev alge-bra as determined by Proposition 3.13, then there exists a nonzeroweight λ of Ao in A with weight space Mλ(α:) = Aa and such that —λis a weight of AQ in A with weight space Λf_λ(—α) = A_α.

We shall identify α with λ as a weight, that is, use the notationa(xQ) for λ(θ50) and also identify Mλ{ά) — Aa, M_λ{—a) — A_«. Notethat Proposition 4.10 implies there exists a basis for A so that for everyx 6 Ao, RX has a matrix of the form

0

0

0

0ra(x)

*

0

o -

a(x)-

0

0

*

0

-a(x)

5 The trace form* Set (x, y) — trace RxRy, then it is shown [3]that this is actually an invariant form-, that is (x, y) is a bilinear formon A such that for all x, yf z e A, (xy, z) = (x, yz). Also a bilinear form(x, y) is nondegenerate on A if (x, y) — 0 for all y e A implies x = 0.

THEOREM 5.1. // A — Ao 0 Aa 0 A_α is α ,/ϊmίβ dimensional simplenon-Lie Malcev algebra over an algebraically closed field of character-istic zero and if A contains an element u such that Ru is not nilpotent,then (x, y) — trace RxRy is a nondegenerate invariant form on A anddimension AΛ = dimension A^a.

Proof. On A = Ao φ AΛ 0 A_a Ru has the matrix

1070

0

0

-a

*

ARTHUR

0

A. SAGLE

0

0

-a 0 Ί

and since u e A = J(A, A, A) (by 1.12) we have by [3; 2.12] that 0 =trace Ru = a(na — n^a) where na — dimension Aaf a — ±a.

Now to show (x, y) is nondegenerate, let T = {x e A: {x, A) = 0}where for subsets J5, C of A we set {B, C) = {(&, c):b e B,c e C) andfor x e A, (x, C) — {{x, c): c e C}. Since (x, y) is an invariant form onA, T is an ideal of A and since A is simple, T — 0 or T = A. If T =A, then (A, A) = 0 and from the matrix of Ru we see that

0 = (u, u) — trace El =

where w = dimension AΛ. Since F is of characteristic zero, a = 0, acontradiction. Thus Γ = 0 which implies (cc, T/) is nondegenerate on A.

COROLLARY 5.2. 7/ A = Ao φ A^ © A_α is α simple Malcev algebraas above then

(Ao, AΛ) = (Λ, A_α) = (Aa, Aa) = (A_Λ, A-*) = 0 .

Proof. Since i?w is nonsingular on Aα, α 0, Aα = Aαi?M. Therefore(Ao, Aα) = (Ao, 4 α K ) = (Aoί2w, Aα) = 0, the second equality uses (a;, j/) isan invariant form and the third uses (2.11). Also (Aα, Aa) = (%Aβ, Aα) =(u, AaAa) c (w, A_α) = 0.

COROLLARY 5.3. / / Ao* is ίfee d^αί s^αcβ of AQ consisting of linearfunctionals on Ao and f e Ao*, then f = ca for some c e F.

Proof. First, (x, y) is nondegenerate on Ao. For if x0 e Ao is suchthat (x0, Ao) = 0, then

\XQ9 Λ.) = (#

c (a?0, Ao) + (x0, Aa) + (xo, A-a)

= 0

by the preceding corollary and therefore x0 — 0 by Theorem 5.1. Nowif / G AQ, then there exists a unique element [2, page 141] af e Ao

such that for all x e A0,f(x) — {x, af) = trace RxRaf —

trace

0

0

0

-a(x)

*

0

o -

a(x)-

0

0

0

*

0 Ί1

0

0

0

0ra(af)

•_ * α

0

o -

(a,)-

0

0

Γ-αία,) 0 -,

= 2na(af)a(x); using the remarks at the end of § 4 to obtain the formof the matrices of Rx and RΛf. Thus / = ca where c = 2na(af) e F.

COROLLARY 5.4. The dimension of Ao is one.

Proof. 0 < dimension AQ = dimension A* = dimension uF = 1.We shall frequently refer to a Malcev algebra A that satisfies

Theorem 5.1 as a "usual simple non-Lie Malcev algebra" and for theremainder of this paper we shall assume the algebraically closed field Fis of characteristic zero.

6 The diagonalization of Ru. Using Proposition 4.10 and Corollary5.4 we are able to decompose A relative to R(A0) into the form

Jx '=- JTLQ (37 -ΛLQJ {37 J\— a

where Ao = uF. From this the matrix of Ru on Aa, a = ±af has theform

-a 0Ί

aJ

We shall show in this section that Ru can be diagonalized. Put Ru intoits Jordan canonical form on Aaf that is, find i?w-invariant subspacesUiia) of Aa such that Aa = U^a) φ 0 Uma(a) and each U^a) has abasis {xil9 , xim.} so that the action of Ru is given by

(6.1)

xiyRu = axix

xi3-Ru = axi5

3 = 2, -

Thus on Z7i(α), Ru has an m x m matrix of the form

a

1 a

1

0

0

1 a

where m = dimension Ui(a). We shall now investigate the multiplicativerelations between the U's and show that the dimension of all the Ui(a)is one and therefore Ru will have a diagonal matrix.

LEMMA 6.2. Ufa) Ufa) = 0.

Proof. Let Ufa) have basis {xlf , xm) as given by (6.1). If m —1, we are finished. Suppose m > 1, then using (1.6)

0 = —J(u, x2, x2)Ru

= J(u, x2, x2Ru)

= α/(w, a?a, a?a) + J(u, x2, xλ)

= J(u, x29 xλ)

— x2xx u + xλu x2 + ux2 a?χ

= x2xλ{Ru - 2α7) .

But we know A2a = 0, therefore sc^a = 0. Now using (1.6) we have, ingeneral, for any i = 1, , m,

0 = J( i6, a?i, XiRJ

= J ( ^ , £Ci, a?<_!) + α J ( w , a?ίf α?,:)

= J(u, xi9 Xi-X)

and again using (1.6),

0 = J(u, xif Xi^Ru)

= J(u, xif Xi-2) .

Continuing this process we have

J{u, xif xk) = 0

for all k ^ i. Now if i < fc, then by the preceding sentence

0 = J{u, xk, Xi) = J ( ^ , aji, a?Λ) .

Thus

J(u, xif xk) = 0 for all i, fc = 1, , m

By linearity this implies

J(u, x, y) = 0 for all a j j e Z7i(α)

Thus

xyRu = Λ?i2M 2/ + -2/K

and

xy(Ru - 2al) = x(Ru - al) y + x y(Ru - al)

As usual we can find an N large enough so that xy(Ru — 2aI)N = 0.But we know A2a = 0, therefore &$/ = 0.

LEMMA 6.3. Lei a? e ^4α δ# s%c/& ίλαί #i?w = α# αwd Zeί Ui(~a) ={l/i, > l/m}» t ^ w ^ i — 0 / o r i = 1, , w — 1 αwcί xym = λu whereX = -(ym,x)l2na.

Proof. Using the invariant form (a?, y) we have (i/wa?, w) = (i/m, a?%) =, »)• Since ^ m e Ao = uF we may write ^ /m = Xu, then (ί/ma?, %) =

(—Xu, u) = — λ(^, w) = — X2na*(a = ±a). Thus λ = —(ym, x)/2na.Now since x e Aa and U^—a) c ^L_α, we have by (2.4) and (2.11) that

0 = J(x, y2, u) = xy2 u + y2u x + ux - y2 — (—ay2 + yjx — α ^ 2 = yxx.Again 0 = J(^, y39 u) = xy3 - u + y3u - x + u% * y3 = (—ay3 + y2)x — axy3 =^2cc. Continuing this process we eventually obtain 0 = J(x, ym,u) =%ym' u + ymu - x + ux - ym = ym-Ύx.

THEOREM 6.4. Let x e Aa be such that xRu = ax and let Ui{—a) besuch that xUi(—a) Φ 0, then dimension Ui(—a) — 1.

Proof. Let B = tfφ # i ^ θ ?/<(—a), then using the precedinglemmas and their notation we see that B is a subalgebra of A and#2/m = xu where λ ^ 0. Now by (2.4) we have J(u, x, ym) — 0, there-fore by [3; Corollary 4.4] we see that u, x and ym are contained in aLie subalgebra, L, of A. However this implies ymu = — aym + τ/w_1 e Land therefore 2/w_! e L; again 2/m_i% = ~aym-x + ?/m_2 e L and therefore7/w_2 G L. Continuing this process we obtain Bzi L and so B is a Liesubalgebra of A Thus for any z e B,

0 = J(z, x, ym)

= z(RxRym — RymRx — RxyJ

- z([Rβ, RyJ - XRU) .

Thus on B we have XRU = [β^, i2 y J and therefore the trace of Ru onJB is zero. But calculating the trace of Ru from its matrix on B, weobtain that the trace is 0 + a — am. Thus m = 1.

COROLLARY 6.5. The dimensional of all the Uii—a), a = ±a, isone.

Proof. Suppose there exists Uj(—a)={yl9 9yn} of dimensionm > 1. Then for every Ui(a)fy1Ui(a) = 0. For if there exists some


Ui(a) such that yJJiia) Φ 0, then by Theorem 6.4, dimension Ui(a) — 1.But this means there exists x e Aa such that xRu = ax and 0 Φxyx G xUά{—α); so again by Theorem 6.4, dimension Uά{—a) — 1, a contra-diction. Thus VίUiia) — 0 for all i and this implies yλAa — y^U^a) 0 0Uma{a)) = 0. Now from Corollary 5.2 we have, since y1 e A_α, (Ao, yλ) =(A_α, I/O = 0 and using the preceding sentence

(Aβ, ί/0 = (Aα, jfctt) = (A,?/!, %) = 0 .

Thus (A, 7/i) = 0 and since (x, y) is nondegenerate on A, yλ — 0, a contra-diction.

7 Proof of the theorem. Let 4 = i o φ i Λ φ A_Λ be the usualsimple non-Lie Malcev algebra, then we have just seen that Aa is thenull space of Ru — al, a = 0, ±a. The choice of a Φ 0 is fixed butarbitrary. In particular we want to consider the case a = — 2, then allwe must do is consider uf = ( — 2/a)u and decompose A relative to Ru,(which is also not nilpotent) to obtain A = Ao 0 A_2 0 Aa. However weshall work with a fixed a and normalize when necessary.

Let α, 6 e F be any characteristic roots (weights) of i?w, that is,α, 6 = 0, ± α with characteristic vectors α?, / e A; that is, αx = xRw by—yRu or x e Aa, y e Ab, then we have

(7.1) J(x, y,u) = xy u — (a + b)xy where x e Aa, y e Ab .

Using (2.4) and (7.1) we also have

(7.2) xy u = (α + 6 ) ^ where y e Aa,y e Ab and a Φb .

Since ## e A_α if α?, 2/ e Aβ, we have

(7.3) xy u = —axy w h e r e x,y e A a .

Combining (7.3) and (7.1) yields

(7.4) J(x, y , u) = —Saxy w h e r e x,y e A a .

Let x,y,ze Aa> then using (2.14), (2.4), (1.9) and (7.4) we have

0 = J(xy, z, u)

= xJ(y, z, u) + J(x, z, u)y —2 J(zu, x, y)

— x{—Zayz) + (—Zaxz)y — 2aJ(z, x, y) .

Therefore

2J(x, y, z) = —3(x yz + xz y)

= 3 {xy z + yz x + zx y) — Sxy z

and thus


(7.5) J(x, y , z) = 3xy z w h e r e x,y,z e A a .

Now J(x, z, y) — Sxz y and adding this to (7.5) yields 0 = xy z + xz yand with a slight change of notation we have

(7.6) xy z = —x yz w h e r e x,y,z e A a .

From (7.6) with z = x we obtain

(7.7) xy x = o where a?, # e Aα .

Now let x, y e Aa, z e A_α, then —aJ(x, y, z) — J(x, y, zu) andJ(z, y, xu) =• αJ(2, 2/, x) = —aJ(x, y, z). So

— 2α«7(aj, i/, z) = J(z, i/, aw) + J(#, y, zu)

= J(z, i/, tt)& + /(a?, ί/, w)« = J(x, y, u)z ,

using (1.7) for the second equality, (2.4) for the third. Thus we have—2aJ(x, y, z) = J(x, y, u)z = (—3axy)z using (7.4) and hence

(7.8) 2J(x, y, z) — Sxy z where x, y e Aa, z e A_α .

This yields 3xy z — 2(xy * z + yz > x + zx y) or

(7.9) xy z = — 2(a?2 y + a? 2/2) where x,y e Aa,z e A_a .

We now use (7.9) to prove the important identity (7.10). Thus letw, x, y, z be elements of Aa and set v = J(x, y, z), 2xf — yz, —2yr = xzand 2s' = a?j/. Then

(7.10) vw = 6(α?'^ a? + 2/'w y + zrw 3) .

To prove this note that x', y\ zf e A_a and using (7.9) we have 2xfx w =xw #' — 2^ίc' #, 2y'i/ w — yw yr — 2wyr 2/, 22;' w = zw 2' — 2wzf 2.Adding these equations and multiplying by 2 yield

2vw = 2(#w #' + yw 7/' + zw 2') + i(xfw a? + y'li; y + z'w z) .

Now using (1.10),

2(xw x' + yw y' + zw z') = xw yz + yw zx + zw xy

= x(zw y) + z(wy x) + w(## z) + #(#2 w) + y(xw 2) + x(wz 7/)

+ w(2ί/ x) + z(yx w) + z(yw a?) + y(wx 2) + w(xz ?/) + α?( ?/ w)

— w(yx ' z) + w(zy x) + w(xz y) + y(xz w) + 2(ya? w) + α?( j/ w)

= —wv + y{—2y'w) + ^(—22;'^) + x{—2xfw)

noting some cancellation to obtain the third equality. Thus 2vw —vw + 2{xrw x + y'w y + z'w 2) + 4(a?'w a? + ί/f/w; y + z'w 2) and thisproves (7.10).


Since A is simple non-Lie Malcev algebra, we shall use the factsA2 = A and A — J(A, A, A) to obtain more identities for A. First wehave

Λ 0 A β 0 A _ β = A = J(A, A, A)

c J(A0, A, A) + J(Aa9 A, A) + J(A_α, A, A)

Am Aa) + J (Ao, A_α, A_α) + J (Aα, AΛ, A^)

_α, A_α, A_α)

and therefore

} Ά—Λ9 Ά — CC)

We now use A = A2 to obtain

Λ Q3 A_α = A — A

and therefore

Ao = AαJA_Q} ,

Since Ao = u F we have A0Aa — Aa(a — ±a). Also

o> A _ α , A _ α ) CI A α

: = Λ.QA.a

d A0«y (Ao, A_α, A__α) + Aoe7 (Aα, A α , A_α)

d e/ (Ao, Ao, A_α) + «/ (Ao, A_α, A—aA.o) -f- e/(^.o, A_α, A 0A_α)

0, Aα, AαA_α) + «/(A0, Aα, A_αAα) + «/(A0, A_α, A | )

obtaining the second inclusion from Aa — /(Ao, A_α, A_α) + J(Aα, Aα, A_o)

and the third inclusion from (1.8). Thus we have

Aa = J(A0, A_α, A_α) , a Φ 0 .

From this and remembering Ao = uF we obtain

Aα = A_aA_a , a Φ 0 .

For A-aA_a (Z Aa = J(A0, A_α, A_α) c A_αA_α. Also

Ao = J(A α , Aα, Aα) , α =

For


Of A.af Λ.a)

ci J\Aa, Ao, Aα) + J(Aa, Aa, AaAQ) + J{Aa, Aa, A0Aa)

d e/(Aα, A.a9 Λ.a) .

We summarize these identities in

PROPOSITION 7.11. Let i = i f l φ 4 0 A-Λ be the usual simple non-Lie Malcev algebra, then we have for a — ±a,

and

" 0 =a, J±a)

THEOREM 7.12. Let 4 = Ao 0 i ^ 0 A_Λ 6e ίfee usual simple non-Lie Malcev algebra, then A is isomorphic to the simple seven dimen-sional Malcev algebra A* discussed in the introduction.

Proof. Since uF = Ao = AJί-Λ — A^ AaAΛ, there exists x,y,ze AΛ

such that % * yz — 2u. Define 2a?' = yz, —2yf — xz and 2z' = #τ/ andform the subspace B generated by {u, x, y, z, xr, yr, z'}. First the x, yand z are linearly independent over F. For if ax + by + cz = 0 witha,b, c e F and, for example, α ^ 0, then write a? = 6'# + c's and there-fore using (7.7) 2u — x > yz = b'y yz + cfz yz — 0, a contradiction.Similarly noting u = $#' and assuming a relation of the type xr —b'y' + c'z' and using the definitions of x\ yf and zr we see that thex\ yr and zf are also linearly independent. Since 4 = 4 O 0 A Λ 0 A_β,{u, x, y, z, x', y\ z'} is a linearly independent set of vectors over F.Using identities (1.2), (7.6) and (7.7) we obtain the following multiplica-tion table for B.

u

X

y

z

X'

y'

z'

u

0

ax

ay

az

—ax'

—ay'

—az'

—ax

0

- 2 2 '

2y'

—u

0

0

y

—ay

22'

0

-2x'

0

—u

0

z

—az

-2y'

2x'

0

0

0

—u

x'

ax'

u

0

0

0

—az

ay

y'

ay'

0

u

0

az

0

—ax

z'

az'

0

0

u

—ay

ax

0

By the remarks at the beginning of this section we can choose a — — 2


and consequently obtain that B is isomorphic to A*. It remains toshow the dimension of A over F is seven. For this it suffices to showdimension Aa = 3, since dimension Aa = dimension A_α. Let 0 Φ w e AΛ,then by (7.5)

6u = 3% * yz = —J(x, y, z)

and therefore by (7.10),

= Gwu = α?ocu + yoί/ + 2:02;

where α?0, i/0, z0 e Ao = uF. But by the action of u on a?, y and s wehave Qaw = αoα; + δ0?/ + coz where α0, δo> o 6 F. Thus the dimension ofAΛ is three.

BIBLIOGRAPHY

1. A. A. Albert, On simple alternative rings, Canadian J. Math., 4 (1952), 129-135.2. N. Jacobson, Lectures in abstract algebra, Vol. 2, D. Van Norstrand.3. A. Sagle, Malcev algebras, Trans. Amer. Math. Soc, 101 (1961), 426-458.

T H E UNIVERSITY OF CHICAGO AND

SRNACUSE UNIVERSITY

MEROMORPHIC FUNCTIONS AND CONFORMAL

METRICS ON RIEMANN SURFACES

LEO SARIO

1. The starting point of the present paper is the classical theoryof meromorphic functions in the plane or the disk. We shall generalizefundamentals of this theory to open Riemann surfaces Ws that carry aspecified conformal metric (Nos. 3, 11). The motivation is that mero-morphic functions are defined by a local property and it is natural toconsider them on the corresponding locally defined carrier, a 2-manifoldwith conformal structure.

The method we shall use largely parallels that of F. Nevanlinna[10] and L. Ahlfors [1]. We have, however, made an effort to writethe presentation self-contained. The classical theory will be includedas a special case.

We note in reference to earlier work generalizations given in variousdirections by L. Ahlfors [2], S. Chern [4, 5], G. af Hallstrom [6],K. Kunugui [8], L. Myrberg [9], L. Sario [14, 15], J. Tamura [19],Y. Tumura [22], and M. Tsuji [21].

2. Our principal result will be the integrated (Nevanlinna) formof the second main theorem on Ws (No. 17). No generalization of thistheorem to Riemann surfaces of arbitrary genus has, to our knowledge,been given thus far. As a corollary the following extension of Picard'stheorem will be established: Let P be the number of Picard values ofa meromorphic function w on a Riemann surface Wp with the capacitymetric (No. 21). Form the characteristic function T(h) of w on theregion Wh bounded by the level line pβ — h of the capacity functionpβ. Denote by E(h) the integrated Euler characteristic of Wh and set

Then

P g 2 + η .

This bound is sharp (No. 27). Analogous extensions will be given toother classical consequences of the second main theorem (Nos. 31-36).

A generalization to arbitrary Riemann surfaces of the nonintegratedform of the second main theorem is given in [18].

Received December 15, 1961. Sponsored by the U. S. Army Research Office (Durham),Grant DA-ARO(D)-31-124-G40, University of California, Los Angeles.

1079

1080 LEO SARIO

§]l. Cotiformal metric

3. Let W be an open Riemann surface. We introduce on W aconformal metric

(1) ds = X(z) \dz\ ,

where X(z) ^ 0 is continuous, with no points of accumulation of its zeros,and ds is invariant under change of parameter z. Other conditions,easily met in our applications, will be imposed upon \(z) in the courseof our reasoning. The length l(a) of a rectifiable arc a on W is welldefined, and the distance d(zlf z2) between two points is inf l{a) forarcs a from zx to z2. The distance d(E19 E2) between two subsets of Wis defined as inf d(z19 z2) for zλ e E19 z2 e E2.

Let Wo be the interior of a compact bordered Riemann surfacecontained in W. We so choose the metric ds that d(z, Wo) tends to aconstant σβ <£ co for any appoach of z to the ideal boundary β of W:

(2) σβ = lim d(zn, WΌ)

for any sequence {zn} tending to β. We consider the ''level lines"

A, = {* I d(z, Wo) = σ} ,

0 g ί/ < σβ, and postulate that ds satisfies the condition

(3) ( ds = 1 .Jβa

Finally, the metric ds is chosen sufficiently regular to justify (3) andother operations to be performed on it. In particular, βσ is assumedto be smooth at points z with λ(z) > 0.

The interesting differential geometric problem of characterizing allmetrics for which these conditions are satisfied will not be discussed inthe present paper. In our applications (nos. 20-87) the conditions aretrivially fulfilled.

4. Schematically, the parameter σ and the arc length s along βσ

constitute a coordinate system on W. If Wσ signifies the relativelycompact region bounded by βσ, then Wσ — Wo corresponds to a rectangleof width σ and height 1 in the (σ, s)-plane. A concrete illustra-tion is given by λ = | grad u\ for a harmonic function u on Wσ — Wo

with u = 0 on β09 u = const, on βσ such that uQdu* = 1. For genus

MEROMORPHIC FUNCTIONS AND CONFORMAL METRICS 1081

g ^ 0 and connectivity c ^ 2 of Wσ — Wo, 2g + 2(c — 2) horizontal slits.appear in the (σ, s)-reetangle; the edges of the slits, and the horizontalsides of the rectangle are suitably identified to form a conformalequivalent of Wσ — Wo. The slits issue from the zeros of Igrad^l.The g ' "handles" of Wσ — WQ give rise to 2g slits in the interior of therectangle, and the c contours cause 2(c — 2) slits terminating at thevertical edges of the rectangle. In the general case of ds = λ | dz | theend points of the slits are at the zeros of λ. The rate of growth of thenumber of these zeros will play a fundamental role in our approach.

§2, The first main theorem

5. Our principal aim is the second main theorem and Picard'stheorem. Since they concern the behavior of a meromorphic functionw on approaching β, it suffices to consider w in the boundary neighbor-hood W— Wo. The first main theorem on arbitrary open Riemannsurfaces will first be needed.

The spherical distance [w, a] between the points w and a is givenby

[w, a] = w — a\l / i + I w | V i + \a

We consider the proximity function

(4) m(σ, a) = — \ log -ds ,2π Jβσ-β0 [w, a\

where the constant l/2π is for convenience in later calculations. Letn(σ, a) be the number of α-points, counted with their multiplicities, ofthe function w in Wσ — Wo. The counting function is defined as

S o-n(σ, a)dσ .

0

For m(*7, oo), n{σ, co), N(σ, oo), the notations m(σ, w), n{σ, w), N(σ, w)will also be used.

6. Differentiation of (4) gives for any α, 6, finite or infinite, andfor σ with no zeros of λ on βσf

dsw — a

/6v dm(σ, a) _ dm(σ, b) = _ l f —\og

dσ dσ 2π )βσ dσ

= — - I d arg w ~ = n(σ, b) - n(σ, a) + nQφ) - nQ(a) ,2π jβ<r w — a

where djdσ stands for the exterior normal derivative in the metricunder consideration, and nQ(a) is the number of α-points in WQ. The

1082 LEO SARIO

differentiation under the integral sign is legitimate, for 1 is an integralJβ<r

with respect to s from 0 to 1. For the characteristic function of wwe choose

(7) T(σ) = m(σ, oo) + N(σ, oo) + n0(co)σ .

On integrating (6) from 0 to σ we arrive at the

FIRST MAIN THEOREM ON RIEMANN SURFACES. Let w be a meromor-phic function on an arbitrary open Riemann surface W. Then

(8) m(σ, a) + N(σ, a) + no(a)σ = T(σ)

for all values a.

We made no use of properties of ds outside of Wσ. In any compactsubregion Ω c W we can choose ds = (l/2ττ) | grs.άp0 I, where pΩ is thecapacity function of Ω = Wσ (L. Ahlfors-L. Sario [3]), and let β0 be alevel line of pβ near its pole in Ω. Thus the first main theorem is ageneral property of w in any compact subregion of an arbitrary W.

7. We observe in passing that the theorem can, of course, also bewritten in the classical form. Let nσ = n + n0, Nσ — N + noσ, andmσ = (l/2τr)( log (l/[w, α])dβ. Then

Jβ<r

(9) mσ(σ, α) + Nσ(σ, a) = T{σ) + 0(1) .

In the case of the 2-plane and for ds — \ dz |/2πτ, this is Nevanlinna's^first main theorem.

8. As is to be expected, the Shimizu-Ahlfors interpretation of the*characteristic function continues to be valid in the present case. Inintegrating (8) over the area elements dω(a) of the α-sphere A theintegral of log [w, α]"1 is independent of w, and the integral of m(σ, a)vanishes. One obtains

(10) T(σ) = — [ ( N(σ, a)dω{a) + Cσ ,π JJΛ

where C is independent of σ.For convenience we shall indicate differentiation by subindices and.

use the notation

dw

(11) |w.| = -ψ- = \w.\ λ-as

Ίz

Then the π^-ίόlά spherical area of the image under w of Wσ — Wo is

S(σ) = -ί (( n(α, απ J J π Jo Jβσ (1 + I w |2)2

and we have

(13) T(σ) = Γsf(σ)dσ + Cσ .Jo

The derivative of the characteristic function T(σ) is, up to an additiveconstant, the spherical area S(σ).

As a corollary one concludes that T(σ) is convex in σ.

§3 Preliminary form of the second main theorem

9. Our next task is to compare the contributions to T(σ) ofm(σ, a) and N(σ, a). To this end we use a mass distribution

<14) dμ(a) = ρ{a)dω(a)

with density p(a) and total mass unity on the sphere A with diameter1 above the w-plane. Again we simply postulate that p(a) is sufficientlyregular to justify subsequent operations on it. This condition will beobviously met by the particular p we shall use.

In the (σ, s)-plane the density takes the form

(15) δ(z) = \J^MI p(w(z)) .* (1 + I w(z) |2)2 rκ v )}

We apply the theorem on the arithmetic and geometric mean toS(z) on βσ:

\ log 3 ds g log I δ ds ,jβσ J Po-

or, equivalently,

(16) ( log —ds +[ logpds^ log ( 8 ds .Jβσ p Jβσ Jβσ

This is the preliminary form of the second main theorem. The proofof the final form consists in evaluating the three terms in (16).

10. The first term depends only on w and λ and will be expressedin terms of T(σ), the counting function N^σ) of the multiple points ofw, and the counting function N(σ, λ"1) of the zeros of λ. The secondterm in (16) depends on the mass distribution dμ. If p is chosen withsuitable singularities at given points αx, « , α g of the w-plane, then

1084 LEO SARIO

\ log p ds will be, in essence, the sum of the proximity functions m(σ, α v ).

In the third term of (16) the integral is the σ-derivative of the mass*on w( Wσ — Wo) and the term will appear as a remainder in the finaLform of (16). Thus the sum ΣιW(σ, αv) will be estimated in terms ofT(σ), Nλ(σ), and N(σ, λ"1). This is the second main theorem on open.Riemann surfaces.

§4. Evaluation of I \og(δlp)ds

11. We set

(17) K(σ) = ±-\ log^-ds = -^\ log \w \ ds4ττ hσ p 2π Jβσ 1 + | w |2

and differentiate:

(18) JP(CJ) = - L A [ log

2π dσ iβ2π dσ her 1 + \w |2 2π iβσ dσ

To evaluate the first integral we have from (4)

m(σ, oo) = — — - I log-- • τds4π Jβσ-β0 1 + \w\2

and consequently

(19) —4-\ log- 1 ^--odm(σf oo)cίσ Jβσ 1 + I w |2 dσ

We now impose upon λ the further condition, always met in ourapplications, that log λ is harmonic except for logarithmic poles. Forthe second integral in (18) the argument principle then gives

(20) -A- ( -A- log I wz\-' I ds - n(σ, -i-Λ - n(σ, w.) -n(σ, -f) + C ,2ττ Jβσ dσ \ wzJ V λ/

where w(σ, λ"1) is the number of zeros of λ in Wσ — Wo, and C is in-dependent of σ. The number of multiple points of w in Wσ — Wo> each&-tuple point counted k — 1 times, is

(21) ^(σ) = w( σ, — ) — n(σ, wz) -\ wz/

and it follows that


K'{σ) = rφ) - 2n(σ, w) - 2

dm^'w^ - n(σ, ±) + C ,dσ V λ/

or, by (8),

(22) K'(σ) = nx{σ) - 2T'(σ) - n(σ, —) +C .

\ x /

On setting

(23) N&) = W(σ)dσ, N(σ, —) = ['n(a, —)dσJo V X / Jo \ λ /

we obtain

(24) -ί- ( log ^-ds = N&) - 2T(σ) - N(G, λ) + Cσ .4π Jβσ p \ xJ

§5 Estimation of \\ogpds

12. Let au a2, * ,αff be g ^ 3 points of the extended w-plane.Choose

(25) - ί log p(w) = ± log _ ! — - log ( t loglog p(w) ± log log ( t log η2 i [w, αv] V i [w, αv]

As ί = [w, αv] —> 0, then ρ(w) —> oo as rapidly as £~~2(log ί)~2, and the mass

\\/>dα> over a ^-neighborhood of αv is dominated by a multiple of

ί"1 (log t)~2dt. Hence the total mass is finite and C in (25) can be

0

chosen to make it unity.

13. Integration of (25) yields

(26) - L ί log p(w) ds = Σ ™(σ> O - 4~ \ log Σ log r

X ds - C ,4ττ J^σ i 2ττ Jj3σ l [ w , α v ]

where

( log ( Σ log Γ -1 Λds ^ log Σm(σ, αv) + C .Jβσ \ i [w, αv] / i

On observing that, by (8), m(σ, αv) ^ 7"(σ), we obtain

(27) -A- f log ^(^)ds ^ Σm(σ, αv) + O(log T(σ)) .47Γ Jβσ 1

1086 LEO SARIO

§6 Estimation of log \δ ds

14. We shall now estimate

(28) ί 8 ds = M'(σ) ,

ho-where M(σ) is the mass distributed on the image of W<r — Wo:

(29) M(σ) = Γ [ δds = (ί ^(σ, α)/o(α)Λϋ(α) .

On setting

(30) Q(σ) = [σΛf(σ)dσ = (ί iV(σ, a)ρ(a)dω(a)

we get from (8)

(31) Q(α) ^ Γ(iί) .

M'{σ) will now be estimated separately in cases σβ — oo and o β < oo.

15. For σβ = co and any constant α ^ 0, let J ' be the set ofvalues σ for which M'(tf) ^ ecύσM{σf% We choose an arbitrarily smallfixed σ0 > 0 and let σ > σ0 in the sequel. Then

j ' ~ Jj' ikf2 M(σ0)

For the set J " of values σ with M(σ) Sg e°iσQ(σ)2 we obtain similarly

f < c o .<Q2

We infer that, for a $ Δ = J ' u ^", ΛP < ^3ασ Q(ί7)4 and consequently

(32) log M'(σ) < Saσ + 4 log Q(σ) .

From (31) it follows that for any a ^ 0

(33) log f δ ds = O(σ + log Γ(σ))

except perhaps in a set J so small that 1 e*σ dσ < co .

16. In the case σβ < co let

Δ' = {σ\ M\σ) ^ eΛl{^~σ) M{σf)

with a > 0. Then

Similarly for

Δ" = {σ I M(σ) ^ e«^~<* Q(σf}

we have

J A" J A" (^

We conclude for σ φ Δ = Δr U 4" that

(33)' log ( δ ds - Of—I + log T(σ)) .)βσ \σβ — σ /

§7* The second main theorem

17. It remains to substitute (24), (27), and (33) or (33') into (16).We have reached the following extension of Nevanlinna's classical the-orem to Riemann surfaces Ws endowed with our conformal metric ds(Nos. 3, 11):

SECOND MAIN THEOREM ON RIEMANN SURFACES. Let w be a mero-morphic function on Ws. For any finite number q ^ 3 of values<x>i> * y Q>q the sum of the proximity functions m(σ, av) grows so slowlythat, if σβ— oo,

(34) Σm((7, O < 2T(σ) - Nτ(σ) + N^σ, —) + O(σ + log T(σ)

except perhaps in a set Δ of intervals with \ ea<x dσ < oo for a ^ 0.J Δ

If σβ < oo, then the term O(σ + log T(σ)) in (34) is replaced by

(34') θ( 1 + logT(σ)\σβ — σ

£/&e resulting inequality holds except perhaps in a set Δ so small

that f β^^-^ dσ < oo for a > 0.

18. An equivalent formulation of (34) is readily found by substi-tuting for m(σ, αv) from (8). For σβ = oo we have

1088 LEO SARIO

(35) (g - 2)T(σ) < Σ>N(σ, αv) - JVi(σ) + N(σ, — + 0(σ + log T(p)) ,

while for σβ < co the term 0(<7 + log T(σ)) is replaced by (34'). Bothinequalities are valid except perhaps in Δ.

19. The presence of exceptional intervals Δ in the second maintheorem was a consequence of the nature of estimation of M'(σ). Sincewe had to start from an upper bound for the integral of the integral

of ikf'(σ), viz., \M(σ)dσ ^ T(σ), a bound cannot always be given for

M\σ) for all σ. If, however, T(σ) and N(σ, λ"1) grow sufficiently

slowly, we shall show that the second main theorem holds without

exceptional intervals Δ.

THEOREM. Suppose T(σ) and N{σ, λ"1) do not grow more rapidlythan e"σ for some a > 0, σβ — co. Then

(36) (q - 2)T(σ) + N^σ) < ΣΛ/Xσ, a*) + N(O, —) + O(σ) .i \ λ /

/ / σβ < co and T(σ) and N(σ, λ"1) are dominated by βc"/(<Γ^~σ) forsome a > 0, then O(σ) in (36) is to be replaced by 0(1 j\σβ — σ)).

Proof. We let N(σ) = ΣiN(σ, av). For σβ = co it follows fromT{σ) -= O(eΛσ) that log T(σ) = O(α ), and (36) holds for σ φ Δ. Now letσ be an arbitrary point of an interval in Δ and denote by σr the rightend point of that interval. Then (36) is true for σr. Since (q — 2)T(σ)+ N^σ) is an increasing function, we have

(37) (q - 2)T(σ) + Nx(σ) < N(σ) + N(σ, j-} + [N(σ') - N(σ)]

From N(σ) = O(eωσ) and the convexity of N(σ) it follows that N'(σ) =eΊσ dσ for Ί>a. By the defin-

σ

ing property of Δf the integral is 0(1). Similarly N{σ\ λ"1) — N(σ, λ"1) =

eΛ(Tdσ, hence σf — σ — 0(1), and weconclude that O(σ') = O(σ). Statement (36) follows.

If σβ < oo, we obtain analogously log T(σ) == O(l/(σβ — σ)) andN'(σ) = Oίe7^^-05^ for some 7 > a. The proof, mutatis mutandis, remainsvalid.

§8* Capacity metric

20. To study consequences of the second main theorem we shallnow leave the above generality of λ and introduce a specific metric.

Let Ω be the interior of a compact bordered subsurface of W, withWo c Ω, and choose a point ζ e Wo. Consider the capacity function pΩ

of the boundary βΩ of Ω. By definition,

(38) pΩ(z) = ^-\og\z-ξ\+h(z)

near ξ in a fixed parametric disk, h(z) being harmonic with h(ζ) = 0.Moreover, pΩ = kΩ = const, on βQ. The functions pΩ form a normalfamily, and any limiting function pβ is a capacity function of β on TFwith pole at ζ [17, 20]. The constant kΩ increases with Ω and tends toa limit kβ oo. The limiting function p β is unique if kβ < cχ>. Thecapacity of the ideal boundary /3 is defined as cβ = e~fc .

For orientation we refer here to two known [3] properties of pβf

although they will not be needed in the sequel: Among all harmonicfunctions p on W with the behavior (38) at ξ, sup^ p is minimized bypβ and the minimum is kβ. The surface W is parabolic if and only ifcβ = 0.

21. We choose the conformal metric

(39) ds = \gτaάpβ\\dz\ .

Set σ = k and denote by βk the level line pβ(z) = fc with 0 k < kβ. Wemay assume that the parametric disk for (38) was so chosen that β0 isan analytic Jordan curve. Then Wo is characterized by pβ(z) < 0, andcondition (2) becomes

(40) lim pβ{z) = kβΩ-+W

with 2 ^ f l . Condition (3), \ ds = 1, is trivially fulfilled. We shall

designate by TΓP a Riemann surface TΓ with property (40) and withmetric (39).

22. Denote by Wk the region pβ(z) < k and consider the Eulercharacteristic

(41) e(k) = —no + v^ — nt

of T7fc — Tfo in a triangulation with ^ 0 vertices, nx edges, and n2 faces.Without loss of generality we may assume that βk consists of a finitenumber of analytic Jordan curves. This can always be achieved by a

1090 LEO SARIO

sufficiently small decrease of k without affecting the subsequent re-asoning.

Our metric (39) has the following property:

LEMMA. The number of zeros in Wk — Wo of grad pβ is the Eulercharacteristic e(k) of Wk — Wo.

The geometric content of the lemma is clear. In fact, the numbern(k, λ"1) of zeros of λ = | grad pβ \ is the same as the number of zerosof the derivative of rj(z) = exp 2π(pβ + ipβ) in Wk — Wo. If βk consistsof one analytic Jordan curve, then the image under rj of Wk — Wo is acircluar annulus with radii 1, e2πk, and tf{z) has no zeros. If βk consistsof two curves, some level line pβ — const, issuing from β0 branches offat a zero z0 of Ύ]\Z) in Wk — Wo to reach the two ft-eurves. If Wk — Wo

is cut along this level line from z0 to βk, the two shores of the cutappear under rj(z) as two radial slits terminating at | η | = e2πk. Moregenerally, if the connectivity of Wk — Wo is c, then there are c — 2zeros of 7)\z) and 2(c — 2) radial slits in the image annulus. A similarreasoning shows that, for positive genus g of Wk — Wo, every handlegives rise to two zeros of η'(z) and two radial slits in the interior ofthe annulus. The total number of zeros of ηf{z) in Wk — Wo is thus

(42) n(k, λ"1) = 2g + c - 2 .

But this is known to be the Euler characteristic e(k) of a bordered surfaceof genus g and connectivity c.

23. To establish our lemma analytically we choose the followingsimple proof given by B. Rodin in his doctoral dissertation [13]. Itshows that the lemma is an immediate consequence of the Riemann-Rochtheorem.

Form the double Wk of Wk by reflecting Wk with respect to βk

([3], p. 119), and denote by g the genus of the closed surface Wk.Extend dpβ + idpβ analytically across βk to Wk so as to obtain ameromorphic differential with two simple poles. By the Rimann-Rochtheorem (e.g. [3], p. 324) the degree of all divisors in the canonicalclass on Wk is 2g — 2. It follows that dpβ + idpβ has 2g zeros in Wk.By symmetry and by our convention in No. 22, g of these zeros are inWk; by our choice of Wo they all are in Wk — Wo. But g = 2g + c — 2,where g and c are the genus and the number of contours of W — Wo.This completes the proof.

24. We introduce the integrated Euler characteristic

(43) Eik) = (* e(k) dkJO

and write our result

(44) n(k, I grad pβ |~0 - e(k)

in the form

(45) N(k, I grader 1) = E(k) .

On substituting this into (35) we obtain the following form of thesecond main theorem:

(46) (q - 2)T(k) < ±N(k, αv) - Nλ(k) + E(k) + O(k + log T(k)) ,1

or, equivalently,

(460 Σm(fc, αv) < 2T(k) - JVi(fc) + E(k) + O(k + log T(k)) .1

Both inequalities hold for kβ = oo, while for kβ < co the term O(& +log Γ(fc)) is to be replaced by O(l/(fcβ - fc) + log Γ(fc)).

§9 Extension of Picard^s theorem

25. We know from No. 8 that T(k) is convex in k. We now ex-clude from our consideration the degenerate case by assuming that

(47) lim

if Jfcβ = oo. By virtue of (13) this means that we only permit functionswith unbounded spherical area S(k) of the image under w(z) ofWk - Wo.

In the case kβ < co we similarly make the assumption

(47') Πm T(k)(kβ - k) = co ,

which implies that S(k) grows more rapidly than ll(kβ — k).An illustrative case is the extended plane punctured at a countable

point set. On this region, despite its weak boundary, there triviallyare meromorphic functions with infinitely many Picard values, e.g., theidentity function. To exclude such functions of no interest we requirethat there be, in some sense, an essential singularity on the idealboundary β. The above condition has this effect:

A meromorphic function with property (47) or (470 comes arbitrarily

1092 LEO SARIO

close to every value a in every boundary neighborhood Wp — Wk.

To see this suppose [w(z)9 a] > ε for all ze Wp — Wko and some a,e, k0 < kβ. Then m(k, a) < (l/2π)logs"1 and N(k, a) ^ n(k0, a)k for k > k0,and we have T(k) = O(&), a contradiction.

26. We set

(48) η =

and denote by P the number of Picard values of w(z). For nondegen-erate meromorphic functions characterized by property (47) or (47') ona Riemann surface Wp we have from (46)

(49) P^2+η- lim

or more simply:

PICARD'S THEOREM ON RIEMANN SURFACES. The number of Picardvalues of w defined on Wp exceeds at most by two the upper limit ofthe integrated Euler characteristic divided by the Nevanlinna charac-teristic:

(50) P ^ 2 + η .

For Riemann surfaces Ws of No. 17 an analogous Picard theoremcan be obtained by replacing k by σ in (47) and (47'), and by substi-tuting N(σ, λ-1) for E(k) in (48).

For functions with E(k)/T(k) —>0 the Picard theorem takes thesimple form P ^ 2 . In particular, this holds for functions on a Riemannsurface of finite Euler characteristic, i.e., of finite genus and a finitenumber of boundary components. In the special case of a plane punc-tured at a finite number of points this is the theorem of G. af Hallstrom[6]. For the nonpunctured plane we have the classical Picard theorem.

27. We claim:

THEOREM. The bound 2 + η for P is sharp.

Specifically, for any integer d ^ 2 there is a Riemann surface Wp

and a meromorphic function w on Wp such that P = 2 + Ύ] = d.

28. For an even d we can make use of the well-known function

w =}* — %

by choosing n = d\2. To this end consider the covering surface W ofthe z = a? + %-plane that consists of ?ι sheets with branch points ofmultiplicity n at z = iπ(i + h),h = 0, ± 1 , ± 2 , . The sheets areattached to each other in the usual manner along the edges of theslits from z — iττ(£ + 2h) to ΐ π ( | + 2h). The function (51) is mero-morphic on W.

To evaluate E(k)/T(k) choose the capacity function pβ — (l/2πn) log \z\on W. It differs from the usual capacity function in that it hasseveral logarithmic poles, one on each of the n sheets above z — 0.However, the behavior of pβ in a boundary neighborhood is unchangedand the reasoning in § 8 remains valid. The set Wo with pβ < 0 consistsof n disks | z \ < 1, but the disconnectedness of Wo has no bearing onour reasoning concerning W — Wo. The metric is ds = | dz \\2πn \ z |,the set βk lies above | z \ = β27rwfc, and the region PΓft lies above \z\ < e2itnk.

In evaluating e(k) and ^(fc, CXD) for rj we may consider TΓ* insteadof Wk — WQ, in view of klT(k)-^0 and of the fact that Wo only con-tributes fixed finite quantities to the'above functions for a given w.

For the Euler characteristic e(k) of Wk we have

(52) e(k) = ne0 (k) + Σbv ,

where eQ(k) is the characteristic of the disk | z \ < e2πnk covered by Wk,and Σbv is the sum of the orders of branch points of Wk. Since eQ(k)= — 1, and I'δv above | z \ < 2π is 4(^ — 1), we obtain on disregardingbounded quantities,

(53) e(k)

Integration from 0 to k yields

(54)

- 1)

•— 1

2ττ

The poles w are the zeros of ez — i, that is, zά — %{π\2 + 2πj) withall integers j . Every pole is simple, and there is only one point of Wk

above each zjt We conclude that

(55)

Consequently

<56)

φ,2ττ

- )

1094 LEO SARIO

From N(k, oo) < T(k) it follows that η ^ 2n - 2. Thus theorem(50) states that the number P of Picard values cannot exceed 2n. Butthis is precisely the number of values w = eίvπln, v = 0, , 2n — 1,uncovered by w(z) (cf. below), and we have proved the sharpness ofbound (50) for even d.

29. A geometric description of the image under w of W with the2n Picard values eίvπln may be illuminating. We first take the ^-sheetedhorizontal strip S of W between y = 0 and y — 2π. It is mapped bys = ez onto an ^-sheeted s-plane Ss slit along the positive real axisand with branch points of order n — 1 at s = ± i . The linear functiont = (s + i)l(s — i) maps Ss onto an ^-sheeted ί-plane St slit along theupper half of 11 \ = 1, and with branch points of order n — 1 at t — 0,co. The function w — t maps S* onto a 1-sheeted region SQ

W of thew-plane less slits L, along | w \ = 1 from β2/A7ri/w to e^+i)«/», ^ = o, 1,w - 1 .

Each ^-sheeted strip 2τrfe ^ y ^ 2π(h + 1) of W is mapped by wonto a duplicate St of SI,. The image Ww of W is obtained by identi-fying the inner edges | w \ = 1 — 0 of all slits Ly on St with the outeredges I w I = 1 + 0 of the corresponding slits L3 on St+1. The processcreates logarithmic branch points at the end points of the slits Lj.The function w has 2n Picard values eiyπln, v — 0, , 2n — 1.

Regarding the sharpness of (50) for odd integers d the reader isreferred to an example constructed by B. Rodin in his doctoral disserta-tion [13].

30. The surface described above has no algebraic branch points,hence Nλ(k) = 0. The question now arises whether or not the bound in(49) can be reached when the surface is so strongly ramified that lim(NjKJήlTik)) > 0. We again use (51) and form the function w = wm,where m is a factor of 2n. A computation similar to the one in Nos.28, 29 yields η S (2n - 2)/ra, hence by (49)

(57) Pm

For m — 1 we again have the bound 2n. Since P and 2njm are integerswe conclude for m > 1 ίnat P cannot exceed 2n/m. But this is clearlythe number of Picard values of w. For any integer q ^ 2 we canchoose n — q and m = 2, say, and obtain q Picard values. We haveshown that the bound in (49) is sharp for all positive integers.

§ 10- Defect and ramification relations

31. We conclude by listing a number of standard consequences of

the second main theorem, extended to meromorphic functions w with(47) or (47') on Riemann surfaces Wp.

The counterpart of the Pίcard-Borel theorem reads: There are atmost 2 + 7] values αv for which lim (N(k, αv)/Γ(&)) = 0.

For the proof we only have to choose q > 2 + η values av in (46)with N(kfav)IT(k)-^0 to arrive at a contradiction.

32. Consider the defect

of w. If ΎJ < oo, then by (46') the number of values a with δ(a) >(η + 2)jn is less than n and one infers that there are only a countablenumber of values a with δ(a) > 0. The following extension of Nevan-linna's defect relation results:

(58) Σδ(a) ^ 2 + η .

33. Let nλ{k, a) be the number of multiple α-points of w in

Wk — Wo, an ί-tuple point being counted i — 1 times. Let Nx{k, a) =

n-JJc, a)dk. The ramification index of a is defined as

lc-+kβ 1 (fC)

It is clear that the set of all multiple points of a given w is countableand that

i ί m ,

T(k) ~ —β T(k)

One obtains the generalization of Nevanlinna's ramification relation:

(59) Σϋ(a) £ 2 + η .

34. Relations (58) and (59) are, of course, special cases of thefollowing consequence of (46'):

(60) Σ8(a) + Σϋ{a) S 2 + η .

For the sum d(a) + &{a) with a given α one has the inequality

(61) δ(a) + ΰ(a) ^ 1 .

This is obtained on dividing

T(k) = m(k, a) + N(k, a) + O(k)

1096 LEO SARIO

by T(k) and on observing that Nx(kf a) S N(k, a).

35. The contribution to T(fc) by the sum m(fc, a) + N^k, a) ismeasured by

θ(a) = lim ^^-

The meaning of θ(a) is clarified by considering the number n(k, a) ofn(k, a)dk

_ o

and note that n(k, a) — n(k, a) — nx(k, a) and N(k, a) = N(k, a) + Nλ(k, a).It follows that

h a) + m(k, a) - T(Je) - N(k, a) + O(k)

and consequently

For the sum of the ^(α) we have the bound

(62) Σθ(a) S 2 + η .

In fact,

Σm(k, a) +^ \[m 2 + ^ .

T{k) ~~ k^rβ T(k) ~~

36. A value a is termed totally ramified if the equation w(z) = ahas no simple roots. The Nevanlίnna relation for totally ramifiedvalues also can be generalized: their number does not exceed 4 + 2f).In fact, for such α, n(k, a) g 2n(k, a). One concludes that

θ(a) ^ 1 - λΈE{N(k, α)/Γ(fc)) ^ 1 .2 fe-fcβ 2

The statement follows from (62).

37. It is an open question whether or not there are functions ona given W with one of the following properties:

(a) P=2 + V,(b) P - 0 but Σδ(a) = 2 + η,(c) Σ#(a) = 2 + V,(d) there are 4 + 2^ totally ramified points.

MEROMORPHIC FUNCTIONS AND CONFORMAL M E T R I C S 1097

BIBLIOGRAPHY

1. L. Ahlfors, ϋber eine Methode in der Theorie der meromorphen Funktionen, Soc. Sci.Fenn. Comm. Phys. Math. VIII. 10 (1935), 14 pp.2. , ϋber die Anwendung differentialgeometrischer Methoden zur Untersuchungvon ϋberlagerungsfiachen, Acta Soc. Sci. Fenn. Nova Ser. A. II. 6 (1937), 17 pp.3. L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton University Press, Princeton,1950, 382 pp.4. S. Chern, Complex analytic mappings of Riemann surfaces I, Amer. J, Math. 82, 2(I960), 323-337.5. , The integrated form of the first main theorem on complex analytic mappingsin several complex variales, Ann. of Math. 7 1 (1960), 536-552.

<6. G. af Hallstrδm, ϋber meromorphe Funktionen mit mehrfach zusammenhdngendenExistenzgebieten, Acta Acad. Aboensis, Math. Phys. XII. 8 (1939), 5-100.7. M. Heins, Riemann surfaces of infinite genus, Ann. of Math. 55, 2 (1952), 296-317.8. K. Kunugui, Sur Vallure dfune function analytique uniforme au voisinage d'unpoint frontiere de son domaine de definition, Japan. J. Math. 18 (1942), 1-39.

*9. L. Myrberg, ϋber meromorphe Funktionen und Kovarianten auf RiemannschenFlachen, Ann. Acad. Sci. Fenn. A. I 244 (1957), 17 pp.10. F. Nevanlinna, ϋber die Anwendung einer Klasse von uniformisierenden Transzen-denten zur Untersuchuny der Wertverteilung analytischer Funktionen, Acta Math. 50•(1927), 159-188.11. R. Nevanlinna, Zur Theorie der meromorphen Funktionen, Acta Math 46 (1925), 1-99.12. •, Eindeutige analytische Funktionen, Springer, Berlin-Gottingen-Heidelberg,1953, 379 pp.13. B. Rodin, Reproducing formulas on Riemann surfaces, Doctoral dissertation, University

•of California, Los Angeles, 1961, 71 pp.14. L. Sario, The second fundamental theorem of meromorphic functions on abstractRiemann surfaces, Contract DA-04-495-ORD-722, Office of Ordnance Research, U. S. Army,Tech. Rep. 18, September, 1960, 16 pp.15. , A Picard-type theorem on open Riemann surfaces, ibid., Tech. Rep. 20,

October, 1960, 17 pp.16. , Meromorphic maps of arbitrary Riemann surfaces, ibid., Tech. Rep. 22,November, 1960, 18 pp.17. , Capacity of the boundary and of a boundary component, Ann. of Math. (2)

.59 (1954), 135-144.18. , Islands and peninsulas on arbitrary Riemann surfaces, Trans. Amer. Math.Soc. (to appear).19. J. Tamura, Meromorphic functions on open Riemann surfaces, Sci. Papers Coll. Gen.Ed. Univ. Tokyo 9, 2 (1959), 175-186.

.20. M. Tsuji, Existence of a potential function with a prescribed singularity on anyRiemann surface, Tόhoku Math. J. (2) 4 (1952), 54-68.

.21. , Theory of meromorphic functions on an open Riemann surface with nullboundary, Nagoya Math. J. 6 (1953), 137-150.22. Y. Tumura, Quelques applications de la theorie de M. Ahlfors, Japan. J. Math. 18(1942), 303-322.

UNIVERSITY OF CALIFORNIA, LOS ANGELES

FACTORIZATION OF POLYNOMIALS OVERFINITE FIELDS

RICHARD G. SWAN

Dickson [1, Ch. V, Th. 38] has given an interesting necessary con-dition for a polynomial over a finite field of odd characteristic to beirreducible. In Theorem 1 below, I will give a generalization of this resultwhich can also be applied to fields of characteristic 2. It also applies toreducible polynomials and gives the number of irreducible factors mod 2.

Applying the theorem to the polynomial xp — 1 gives a simple proofof the quadratic reciprocity theorem. Since there is some interest intrinomial equations over finite fields, e.g. [2], [4], I will also apply thetheorem to trinomials and so determine the parity of the number ofirreducible factors.

1. The discriminant* If f{x) is a polynomial over a field F, thediscriminant of f(x) is defined to be D(f) = δ(/)2 with

where alf , an are the roots of f(x) (counted with multiplicity) insome extension field of F. Clearly D(/) = 0 if / has any repeatedτoot. Since D(f) is a symmetric function in the roots of /, D(f) e F.

An alternative formula for D(f) which is sometimes useful may beobtained as follows:

D(f) = Π(«i - <*sY = (-1)-'—1>/aΠ(«i - «;) = ( - i r ' - ^ ' Π / ' ί α O

where n is the degree of f(x) and f\x) the derivative of f(x). In § 4,I will give still another way to calculate D(f).

If f(x) is monic with integral coefficients in some p-adic or alge-braic number field, all at are integral and so D(f) is integral. Considerthe expression

This is integral and lies in F, being a symmetric function of the roots.Clearly δ(/) = 8X + 2δ2 where δ2 is integral. Thus D(f) = δ(/)2 = dl mod4, so D(f) is congruent to a square in F mod 4. This is a special caseof a well-known theorem of Stickelberger [3, Ch. 10, Sec. 3].

Received November 15, 1961. The author is an Alfred P. Sloan Fellow.Added in Proof. I have recently discovered that Theorem 1 of this paper is due to

L. Stickelberger, ϋber eine neue Eigensehaft der Dίskrimίnanten algebraischer Zahlkorper,Verh. 1 Internat. Math. Kongresses, Zurich 1897, Leipzig 1898, 182-193. A simplified proof,•essentially the same as mine, was given by K. Dalen, On a theorem of Stickelberger, Math.Scand. 3 (1955), 124-126.

The applications of the theorem, however, seem to be new.

1099

1100 RICHARD G. SWAN

Suppose f(x) is monic with integral coefficients in a p-adic field F~I will denote by f(x) the polynomial over the residue class field obtainedby reducing all coefficients of F mod p. In some extension field of Fwe have f(x) = (x — ax) (x — an). Therefore f{x) = (x — ax) (x— an) where α is ai reduced modulo the (unique) extension of p. Itfollows that D(f) is D(f) reduced mod p. In particular, if f(x) has-no repeated root, D(f) Φ 0 and so D(f) is prime to p.

2 The main theorem* If f(x) is a monic polynomial with integralcoefficients in a t>-adie field, I will again denote by f(x) the result ofreducing the coefficients of f(x) mod p.

THEOREM 1. Let f{x) be a monic polynomial of degree n withintegral coefficients in a p-adic field F. Assume that f(x) has norepeated roots. Let r be the number of irreducible factors of f(x) overthe residue class field. Then r = n mod 2 if and only if D{f) is asquare in F.

Proof. Over the residue class field K we can factor f(x) = f^x}• /r(#). Since the /<(#) are relatively prime, HenseΓs lemma showsthat there is a corresponding factorization f(x) = fx(x) fr(r) over Fwhere f{(x) is fi(x) mod p. Since f^x) is irreducible over K, fi(x) isirreducible over F. Since /<(») has no repeated root, D(fi) is prime to^/. Therefore the field Et obtained by adjoining a root of f{(x) to Fis unramified over F. Since there is only one unramified extension ofF of any given degree and that extension is cyclic, E{ is cyclic over Fand thus is the splitting field of f{. The splitting field E of f(x) is thecomposite of all the E{ and therefore is unramified over F. Thus E/F'is a cyclic extension. (A more elementary proof of this was pointedout by W. Feit. Let m be the least common multiple of the degreesof the fi(x). It is easy to construct a cyclic unramified extension EJF1

of degree m by adjoining a root of unity to F. Now f(x) splits com-pletely over the residue class field Kx of Eλ since the degree n{ of everyirreducible factor of f(x) divides m = [Ex\ F], By HenseΓs lemma, f(x}splits completely over Ex so E c Ex.)

Let σ generate the galois group of E/F. Let β5 be a root of f,(x).Then the roots to f3(x) are given by σι{β3) f or 0 ^ i ^ n5 — 1 where n3-is the degree of f,(x). Thus the roots of f(x) are (/Sy) fori = 1, •••,n, i = 0, , n3- — 1. These can be ordered by defining (ilf j\) < (ia, i2)if oΊ < U2 or if o\ = i2 and ^ < ia. For any ΐ, the symbol (i, j) will beinterpreted to mean (ΐ', j) where V = i mod %, 0 ^ if ^ % — 1.

Now D(f) = δ(f)2 where

= Π {σ^βh -

FACTORIZATION OF POLYNOMIALS OVER FINITE FIELDS 1101

If we apply σ to <?(/), we get

σδ(f) = Π (

The terms occurring in this product are the same as those in <?(/)except for sign. The term σίl+1/5ii — σi2+1βJ2 occurs in d(f) and oδ(f)with the same sign if and only if (it + 1, j\) < (i2 + 1, j2). This is cer-tainly the case if j \ < j2 or if j \ = j2 = j and i2 < nj — I. However, ifii = j \ = i and ia = % — 1, then (ia + 1, j) = (0, i) < (ix + 1, j). There-fore the total number of terms which occur with different signs in <?(/)and σδ(f) is equal to the number of pairs ((ilf j), {nό — 1, j)) where0 ^ ii ^ ^i - 2. There are n$ — 1 such pairs for each j . The totalnumber is given by

2 ( % — 1) = n — r .3

This shows that σδ(f) = (-l)%-r<5(/). Now D(f) is a square in F ifand only if δ(f)eF, which is the case if and only if σδ(f) = δ(f).Therefore D(f) is a square in F if and only if n = r mod 2.

COROLLARY 1. Lei K be a finite field of odd characteristic. Letg(x) be a polynomial over K of degree n with no repeated root. Letr be the number of irreducible factors of g(x) over K. Then r = nmod 2 if and only if D(g) is a square in K.

Proof. We can assume that g(x) is monic. Let F be a p-adic fieldwith residue class field K. Let f(x) be a monic polynomial over F withintegral coefficients such that f(x) = g(x). Then D(g) is D(f) reducedmod p. Since D(g) Φ 0, D(f) is a square in F if and only if D(g) isa square in if. This follows immediately from HenseΓs lemma appliedto the polynomial x2 — D(f).

A more elementary proof of Corollory 1 can be obtained by repea-ting the proof of Theorem 1 using K in place of F.

If f(x) is irreducible over K, r = 1 and so n is odd if and only ifD(f) is a square in K. This is the theorem of Dickson mentionedabove.

If K has characteristic 2, the proof of Corollary 1 breaks downbecause D(g) may be a square mod p even though D(f) is not a square.For example, 3 is a square mod 2 but not mod 8. In this case weneed the following well-known result.

LEMMA 1. Let a be a p-adic integer prime to p. Then a is ap-adic square if and only if a is a square mod 4p.

Proof. S u p p o s e a = b\ m o d 4ψ\ T h e n a = b\ + Ac{ w h e r e C { Ξ 0

mod pι. Let d{ = bi%. Then ^ = 0 mod p* since 6< is prime to p,and a = (bi + 2dtf - 4d?. Let bi+1 = bt + 2c^. Then α = 62

ί+1 mod 4jjt+i ^ j n fac . m 0 ( j 4p2i^ rpjie ^ form a Cauchy sequence and a = b2

where 6 = lim 6 .

COROLLARY 2. Lei /(a?) δβ a monic polynomial of degree n withintegral coefficients over a p-adic field F. Assume that f(x) has norepeated root. Let r be the number of irreducible factors of f(x) overthe residue class field K of F. Then r ΞΞ n mod 2 if and only ifD(f) is a square mod 4£.

This follows immediately from Theorem 1 and Lemma 1. In apply-ing Corollary 2 we are usually given K and f(x) and choose any con-venient F and f(x). For example, in case K — GF{2), we get

COROLLARY 3. Let g(x) be a polynomial of degree n over GF(2)with no repeated root. Let r be the number of irreducible factors ofg(x) over GF(2). Let f{x) be any monic polynomial over the 2-adicintegers such that f(x) — g(x). Then r = n mod 2 if and only ifD(f) = 1 mod 8.

This follows immediately from Corollary 2 and the fact that 1 isthe only odd square mod 8. Note that D(f) = 1 or 5 mod 8 by Stickel-berger's theorem.

EXAMPLE. Let f(x) be a polynomial of degree k over a finite fieldof characteristic 2 such that /(0) φ 0. Let g(x) = f(xf + xm where mis odd. Then n = deg g(x) — max (8k, m). Choose an appropriate p-adic field and a polynomial F(x) of degree k such that f(x) = F{x).Then g(x) = G(x) where G(x) = F(x)8 + xm. Now G'{x) = mx™-1 mod8 so

D(G) = (-ly^^Umaf-1 (mod 8) .

Since Πa, = (-l)nG(0) Ξ /(O)8 0 mod p, D(G) Ξ£ 0 mod ί> so flf(α?) hasno repeated root. Since m is odd, Πaf~x is a square. Thus D(G) differsby a square factor from

D' = (- l) <»-«/ϊw .

lί n — 8k, D' is a square. Thus r = n = 0 mod 2. Therefore #(#) hasan even number of factors and so is reducible. If n — m, Dr differsby a square factor from (-l)(m"1)/2m. If m = ± 3 mod 8, (-l)(m-1)/2m Ξ 5

mod 8 and so r -φ n = 1 mod 2. Therefore g{x) is reducible if m Ξ ± 3mod 8.

In particular, xsk + xm + 1 is reducible mod 2 if m < 8ft. If m > 8ftit is reducible if m = ± 3 mod 8.

3. Quadratic reciprocity• The discriminant of the polynomial xn + aover any field is given by

nf.n—1D(xn + a) = (—1)*CΛ-1)/2 Π^^?" 1 = (-l)n{n-1)l2nna

because 77^ = (—l)wα.Consider in particular the polynomial xp — 1 = (x — l)Φp(x) where

φ is an odd prime. Its discriminant differs by a square factor from<—l){p~1)l2p. Therefore xp — 1 has an odd number of factors over GF(2)if and only if p = ± 1 mod 8. If q Φ p is an odd prime, xp — 1 hasan odd number of factors over GF(q) if and only if ( — l){p~1)βp is asquare mod q.

Now, if α is any root of Φp(x) over GF{q), q ψ p, a is a primitivepth root of unity. Therefore α e GF(qn) if and only if p | qn — 1. Thusthe degree of a (and hence of any irreducible factor of Φv{x)) overGF(q) is the order n of g mod p. It follows that xp — 1 has 1 + φ(p)/nfactors over GF(q)p

Since the multiplicative group Z* of integers mod p is cyclic, Z*has a unique subgroup of index 2 which consists of all squares. Thusq is a square mod p if and only if it generates a subgroup of Z* ofeven index. This index, however, is just φ{p)jn, so q is a square modp if and only if xp — 1 has an odd number of factors over GF(q).Comparing this with the results obtained from Theorem 1, we get

i) = ( H p ) « „ is

—) = 1 if and only if p = ±1 mod 8 .pJ

These equations, together with the trivial formula

— 1 \(p-D/2

p

constitute the quadratic reciprocity theorem.

4. Calculations* For the calculations made above, the discriminantcould easily be found using the formula given in § 1 . However, formore complicated polynomials this method of finding the discriminant isvery inefficient. In this section I will give a simpler method based on


the Euclidean algorithm and use it to calculate the discriminant of anytrinomial.

Let / and g be any polynomials over any field F. Let g have rootsβi, m,βm, (counted with multiplicity), and leading coefficient 6. Letn be the degree of f(x). Then the resultant of / and g is definedto be

R(f, 9) - ^Uf(βj) .3=1

Clearly R(f, g)e F since it is a symmetric function in the roots of g~Comparing this definition with the formula for D{f) given in § 1 showsthat if / is monic

The following properties of R(f, g) are presumably well known, butI will include them for completeness.

LEMMA 2. (1) R(g,f) = (-l) d e^ d^Λ(/, g)(2) Iff = gq + r,

R(f, g) = &**/-**'jβ(r, g)

where b is the leading coefficient of g.(3) If a and b are constants not both 0, R(a, b) = 1.

(4) Λ(Λfaf g) = B(f19 g)R(f» g) .The proof is trivial.

COROLLARY 4. (5) R(f, gxg2) = R(f, gi)R(f, g2)(6) If a is contant, R(f, a) = aάesf - R(a,f)(7) Λ(/faj-) = 22(/f*) =/(O) .It follows from properties (1), (2), and (3) of Lemma 2 that we

can compute R(f, g) by applying the Euclidean algorithm to / and g.This method of computation seems much easier in practice than therather cumbersome determinant formula [5, Ch. 11, §71, 72.].

In order to compute the discriminant of a trinomial, it is firstnecessary to compute the resultant of two binomials.

LEMMA 3. Let d = (r, s) be the greatest common divisor of r ands. Let r = dr19 s = ds,. Then R(xr - a, xs - β) = (-l)s[asi - βrψ.

Proof. We first observe that if the result holds for a given pair(r, s) it holds for (s, r). This follows easily from Lemma 2 (1) usingthe fact rs + s + d = r mod 2.


Since the result is trivial for s = 0, we can prove it by inductionon r + s, assuming also r ^ s by the previous remark.

Now, dividing xr — a by xs — β gives the remainder βxr~s — a.Thus we can apply Lemma 2 (2) and the result follows easily by in-duction.

It is now easy to find the discriminant of a trinomial over anyfield.

THEOREM 2. Let n > k > 0. Let d = (n, k) and n = ^d, & = i eZ.

D(xn + α#* + 6) = (-l)*t

Proof. Consider the generic polynomial / of degree n, multiply out

D(f) = Π(«i - a;)2

and express the resulting symmetric functions as integral polynomialsin the coefficients of /. This gives an expression for D(f) as a specificintegral polynomial in the coefficients of / and this expression is in-dependent of the characteristic. In order to find the form of thispolynomial, it is sufficient to do it in characteristic 0. For any poly-nomial / of degree n,

Therefore

D(xn + axk + b) = (-ly^-WRinx*-1 + kaxk~\ xn + axk + b)

= ( — \y^-^2nnbh"1R{xn + axk + δ, xn~k + n

using Lemma 2 (1), (4) and Corollary 4 (7)Now, dividing xn + axk + δ by xn~k + n~τka leaves the remainder

α(l — n~λk)xk + δ. Therefore the result follows from Lemma 2 (2) andLemma 3.

As an application of Theorem 3, I will determine the parity of thenumber of factors of xn + xk + 1 over GF(2).

COROLLARY 5. Let n > k > 0. Assume exactly one of n, k is odd.Then xn + xk + 1 has an even number of factors {and hence is re-ducible) over GF(2) in the following cases.

(a) n is even, k is odd, n Φ 2k and nk\2 = 0 or 1 mod 4(b) n is odd, k is even, k \ 2n, and n = ± 3 mod 8(c) n is odd, k is even, k | 2n, and n == ± 1 mod 8

In all other cases xn + x + 1 has an odd number of factors over GF(2).


The case where n and k are both odd can be reduced to the casek even by considering xn + xn~k + 1 which has the same number of ir-reducible factors as xn + xk + 1.

To prove Corollary 5 we regard xn + xk + 1 as a polynomial overthe 2-adic integers, compute its discriminant by Theorem 2, and applyCorollary 3.

Note that the fact that some trinomial xn + xk + 1 has an oddnumber of factors does not imply that it is irreducible. For example,we may consider the trinomial x2k + xk + 1 with k odd. In a numberof cases, including this one, the reducibility or irreducibility of xn + xk

+ 1 can be decided by using the results of [1, Ch. V, §9]. Recall thatan irreducible polynomial f(x) over a finite field is said to belong to theexponent e if e is the least integer such that f(x) | xe — 1. In otherwords, e is the order of a root of f(x).

If f(x) is irreducible of degree n and exponent e over GF{q), itfollows from [1, Ch. V, Th. 18] that f{x8) is irreducible over GF(q) ifand only if every prime p dividing s also divides e but does not divide(q* — l)/β and, in addition, 4 | s if qn ΞΞ —1 mod 4.

In particular x2k + xk + 1 is irreducible over GF(2) if and only ifk is a power of 3 and x4k + xk + 1 is irreducible over GF(2) if andonly if k = 3r5s.

Some other cases can be disposed of by observing that if xr + xs + 1divides xβ + 1 then xr + xs + 1 divides xn + xk + 1 if n = r, k = s mode. For emample, x2 + x + 1 divides xn + x& + 1 if n = 2, fc = 1 mod 3or if ^ Ξ 1, & = 2 mod 3.

REMARK. I. Kaplansky points out that Theorem 1 can be refor-mulated so as to avoid the use of p-adic numbers by considering thepositive and negative terms in Π{ai — a;). These are polynomials inthe ai which also make sense in characteristic 2. This yields an ele-mentary form of the theorem but one which is hard to apply becauseof the difficulty in computation.

REFERENCES

1. A. A. Albert, Fundamental Concepts of Higher Algebra, University of Chicago Press,1956.2. , On certain trinomial equations in finite fields, Ann. of Math., 6Θ (1957),(1957), 170-178.3. E. Artin, Theory of Algebraic Numbers, Gottingen, 1959.4. H. S. Vandiver, On trinomial equations in a finite field, Proc. Nat. Acad. Sci. U.S.A.4 0 (1954), 1008-1010.5. B. L. van der Waerden, Moderne Algebra, Julius Springer, Berlin, 1931.

T H E UNIVERSITY OF CHICAGO

SOME THEOREMS ON THE RATIO OF EMPIRICALDISTRIBUTION TO THE THEORETICAL

DISTRIBUTION

S. C. TANG

1. Introduction. Let Xlt X2, " ,Xn be mutually independent ran-dom variables with the common cumulative distribution function F(x).Let X*,X*, "',X* be the same set of variables rearranged in in-creasing order of magnitude. In statistical language Xu X,, , Xn

form a sample of n drawn from the distribution with distribution func-tion F(x). The empirical distribution of the sample Xu •• ,Xn is thestep function Fn{%) denned by

( 1 ) F,(x) =

0 for x ^ X*

— for Xξ < x 5ϊ Xk*+1it

1 f or x > Xί .

A. Kolmogorov developed a well-known limit distribution law forthe difference between the empirical distribution and the correspondingtheoretical distribution, assuming F(x) continuous:

(0 , for z ^ 0 ,

^P{Vn-^?JFM ~ F^< 4 = {ti-iye— , for * > 0 .

Equally interesting is Smirnov's theorem:

lim p\VH sup [Fn(x) - F(x)] < z\ = ]° ' __2β2

0 , f or z ^ 0 ,

f or z > 0 .

In this paper we shall study the ratio of the empirical distributionto the theoretical distribution, and evaluate the distribution functionsof the upper and lower bounds of the ratio. We shall prove the follow-ing four theorems.

THEOREM 1. If F is everywhere continuous then

(0 for z%l

— JL for z > 1 .z

Received July 28, 1961. I am deeply indebted to the referee for his corrections of myEnglish writing. Without his aid, the present paper would not be read clearly, althoughthe responsibility of errors rests with me.

Π07

1108 S. C. TANG

THEOREM 2. Let c be a positive number no larger than n andsuppose that F(x) is continuous in the range 0 ^ F(x) g c/n. Then

ίj. F(X)

£? (n\ kk-\nz - &)—*~~ 2-*\k) 7 ^

Λ=I \κ/ (nz)

m ίg /w - fey ca - k γ~*Λ _ eg - fe y«•—u \ ι ~~~ iC / \ /yi/y ^ _ If / \ 'V?<y — i* /i —K \ / \ /t, /v / \ / (ί/& /v /

\

THEOREM 3. Under the assumption of Theorem 1,

for 0 < z S

for z ^ 0

for z> —c

—e

\ i n f Ή*lβ<#.(.)si F(X)

for z < —n

1 -nz

(fe - l ) * - 1 ^ - fc)""fe

/or 1- ^ ^ <n

for z > 1 .

THEOREM 4. Under the assumptions of Theorem 2, if 1 <Ξ c : nthen

P^ inf

0

Λ _ 1V nzJ

/ o r ^ < JL

(fe - l ) * - 1 ^ - k)n~*

(nz)n

for — ^n

—n

v n\ (fe

forz>j~.

2. An elementary lemma. We shall use the following lemma.

SOME THEOREMS ON THE RATIO OF EMPIRICAL DISTRIBUTION 1109

LEMMA. Let fc be a positive integer and a an arbitrary real num-ber. Then

(2) sp r [a r) afϋ. r! ( fc-r) ! (fc - 1)! '

(a - 1)* ^ (r - I ) - 1 (a - r)«-' _ a*k\ ί=i r! (fc-r)l fc! '

If a = k (a special ease) then

1 } h r! (fc - r)! (fc - 1)! fc! '

,,, (k - I)" ^ (r - I ) - 1 (k - r)*-r _ k«κ ' M #=i r! (fc-r)I fc! '

Proof. Let /(«) be a polynomial whose degree is less than &. Then

i) r/(* ) = (-D"[^/(*)].=. = o

i.e. Σ(ί)(-l)r/(r)=-/(0).

JSΓow we can directly obtain (2) and (3):

(α-Dfc!

r\

k_L.

1J

(a

(a

(a

t o

( f c -

-\yfc!

-D*fc!_ 1 ) »

fc!(a-

s! (fc

r)\

r\

- 4-

- +

— s

z^ -

( -

y-1 («-(fc

^ ( r -

r\ s=o s! (fc

-f \ 1. Λ g λ. g

i.^ Oί s^l

S! r=i

.l)*-«-«α*-i

(fc - 1)!

- r ) !

1 ) ^ ^ (α -r=i 7*' s=0 ,

v 1 (-1 's = 0

^(-1)

)! fc!

)*-=(« _ i ) .

s!

*-( α _ i) (_s! (fc - s)!

— r —

r!(fc-

( 1)-

- l ) s ( -

s ! ( f c -

^ ( - •

fA. r!

β ) !

- r —

(fc

-r+r —

L)'(r

( f c -

-1

•s)l

a"-1

- 1)! '

\y-r-*

β ) !

r — s)!

3. Proof of Theorem l First we consider the case z 1. Letbe the largest root of the equation

1110 S. C. TANG

F(x) = A (i ^ k n) .nz

Since F is continuous, yk is well defined. Now we evaluate prob-ability of the inequality

(6) sup ΣΆ0^F()^ F(X)

that is, the probability that there exists an x such that

(7) F ^ > z .F(x) ~

If (7) is true, then, since F{x) is nondecreasing and lima._0o (Fn(x)IF(x)) —1, there exists an xQ such that

F(xQ)

By the definition of Fn, Fn(xQ) = kjn for some k, so that F(xQ) —kfnz. Therefore we can take x0 as one of yk. In other words, for onevalue of yk we have

FM) = -n

and the inequality

(8) Xt<y*^Xί+i

is true. Now let Ak be the event that this inequality holds.Clearly (6) occurs if, and only if, at least one among the events

A A A . . . A

occurs. Generally the Afc are not mutually exclusive events so that theadditive law is of no use. We may, with the help of an associated setof mutually exclusive events, deal with this situation. Put

uk = AXA2 Ak^Ak {k = 1, 2, , n) .

where A denotes the complementary event of A. We have

(9) P{ sup 1M- ^z} = Pl±Ak\ = Pi±uλ = ΣPW .U<F{χ)^i Fyx) ) U=i J U=i J fc=i

If we employ the following conditional probability formula, we canevaluate P(uk).

(10) P(Ak) =r=l

Now Ak occurs if k of the n observations fall to the left of yk,and n — k to the right; hence

nz / \ nz

And P(Ak I Ar) is the probability that of (n — r) observations, known tolie to the right of yr, (k — r) lie to the left of yk and n — k to theright. The probability of occurrence of an event, whose value is greaterthan or equal to yr and is on the interval yr^x ^ yk is

kjnz — r\nz _ k — r

1 — rjnz nz — r

Therefore

k - r \*-V1 _ k- r Y~k

z — r / V nz — r

If we employ the notation

pk(z) =kl

we get

Equation (10) can be reduced to

kJz) _ Λ p( v pk-r(l)pn-k(z)

= Σ PK)

Hence P,(l) = fcfc/^! and P,_r(l) = (k - r)*-r/(fc - r)!. By (3) of ourlemma we know that

P(Mr) P»(g) Z ! l i ΪHL P ( ) 1 1 1 fo ) " r

p ( ^ ) r! ' r r! pn(«) (nz)n rl (n — r)\

And (9) and the (2) of our lemma tell us that

P{ sup ^ L < z) = 1 - P{ sup -§M. ^

{nzf έΊ fc! (n - k)\

- 1)!

1112 S. C. TANG

This completes the proof of Theorem 1.

Proof of Theorem 2. Since the ratio is nonnegative, the prob-ability of the inequality is zero if z ^ 0. Under the condition z > njc,we know that the event {supo<r{x)^eιnFn(x)IF(x) Ξ> z} is still equal toΣt=i A by the result of Theorem 1. Suppose 0 < z ^njc. Then

P i QITΠ n\fi) > ? I — p j sf* A J_ A * I — 'S? PΓ?/ ^ 4- T^di^Λ

lθ<F(x)^ F(X) ) U=l J fc=l

where

{ c 1a?: F(^) = —^ ,

n J1/* - 4 J . , , J 4*

Since P{Uk) has been computed, we need only evaluate P(C/c*). Wehave

P(A*) - Σ P(uk)P{A* | A,} + P(u*)P{A* \ A*} ,

P(u*) = P(A*) - Σ P{A* | Ak} .

From this we obtain

pi gup ^ ί ί L < z\ = 1 - Σ P(^fc) - P(%*)lo^ί'(χ)^ F(x) > k=i

_ . v pίηi \\Λ P / 4 * I A W

ίczl

JL \ wkJ-L χΛ~\.C2 £±-k\

Since

r = 0

we have

γ Ί _ ^ - f e y -

which completes the proof. The distribution of Theorem 1 is continuous;the distribution of Theorem 2 is not continuous on the interval 0 ^z gΞ n/c.

Theorem 3 is a special case of Theorem 4 under the condition c = n.In fact, by (4) of our lemma, setting z = 1, we deduce Theorem 3 asfollows:

{ ininf ^ < i U ( i - λ)+ ± (I)F(x) ~ ) \ n) * = i W nn

n L n\ lέi fc! (n - fc)! J

Thus we need only prove Theorem 4. The distribution function ofTheorem 3 has a discontinuity corresponding to 3 = 1; the distributionfunction of Theorem 4 is continuous on the interval 1 c ^ n.

5. Proof of Theorem 4 On the internal Fn(x) > 0 the maximumof F(x) is 1; the minimum of Fn(x) is 1/n. From these results it fol-lows that the lower bound of the ratio is no smaller than 1/n. There-fore

w*i F(x)

if z < ljn. Let zk be the least root of the equation

nz \ n J

Define events

J30. A x ^ Z1 ,

If 1/n z ^ c/w, the event

is the union of the events

Bo> Bi> B2f , Bίnzl .

For the purpose of evaluating the probability of ^kBk we put

Vo - Bo, Vk= SoB1... S t _ A (fc = 1, 2, . . . , [nz])

1114 S. C. TANG

We have

(11) P(Bk) = Σ P ( Vr)P(Bk \Br), (fc = 0,1, , [nz]) „

Here

F{Bk \jbo\ —

Now we transform (11) into the following form:

klpn(z) r=i r pn_r(z)

By (4) of our lemma we obtain

P(Fr) = ( r ~ 1>r'1 p*-*iz) = (n)(r-lY-K™-'> ( 1 < r <r! pn(z) \r/ '--sn ~ ~

It follows that if 1/n g z ^ c/w,

i \ _1_ "S? I \ N — / \ιvZ — K>)

nz / k=i\K'/ (nz)n

If z > c/w,{ inf ^U<Fn{x)^~ F(X)

P{ ^ ^ }U<Fn{x)^~ F(X) >

= Λ _ M

Theorem 4 is thus proved.

6 Conclusions, Theorems 1 and 3 can be applied to test whetherstatistical data correspond with the theoretical distribution or not. Inthe process of testing, zs and z{ are separately representative of theratio's upper and lower bounds. If S(zs) is small and In{z^) large, weconclude that the empirical data in two extreme tails correspond withthe theoretical distribution; conversely, if S(zs) is very large and In{z^very small, the statistical data do not correspond with the theoreticaldistribution. Our test is more sensitive in the two tails, but less sensi-tive in the central part, than Smirnov's test.

NORMAL MATRICES AND THE NORMAL BASISIN ABELIAN NUMBER FIELDS

R. C. THOMPSON

l Introduction. Throughout this note F denotes a normal field ofalgebraic numbers of finite degree n over the rational number field.Let Glf G2, •••,£» denote the elements of the Galois group G of F. Itis known [2] that F may possess a "normal" basis for the integersconsisting of the conjugates aG\aG\ • • ,αG^ of an integer α. In [4]the question of the uniqueness of the normal basis was answered whenG is cyclic. (See also [1, 6].) If βl9 β2, •••,£» is any integral basis ofF then the matrix (βp), 1 < i, j ^ n, is called a discriminant matrix.It was shown in [4] that if G is abelian then the discriminant matrixof the normal basis βx = aθί, , βn = αff» is a normal matrix and, if Gis cyclic and F has a normal basis, then any integral basis βlf , βn

for which the discriminant matrix is normal is of the form βσ{1) =±a&1, •• ,ArU) = ±oP» for a suitable choice of the ± signs, where σis a permutation of 1,2, , n.

It is the purpose of this note to use the methods of [4] to extendthese results for cyclic fields to abelian fields. In particular, in Theorem1, we shall give a new proof of a result obtained by G. Higman in[1]. The author wishes to thank Dr. 0. Taussky-Todd for drawing theproblems considered here to his attention.

2. Preliminary material* We suppose throughout that

x(Sk)

is the direct product of k cyclic groups (Si) of order nζ. Of course,each rii > 1 and n = nxn2 nk. If X and Y = (yitd) are two matriceswith elements in a group or a ring then we define X x Y = (Xy^).I x F is the Kronecker product [3] of X and Y. Henceforth, in thispaper, the symbol x will always be used to denote the Kroneckerproduct of vectors or matrices. A matrix A is said to be a circulant•of type (n±) if

a 2 α 3 ••• a n i

(Xi (Xo Ct«,_

A = [(&!, α 2, , α n J W l =

Ήere al9 a2, , αΛl may lie in a group or a ring. For i > 1 we define

Received June 15, 1961, and in revised form October 13, 1961.

1115

1116 R. C. THOMPSON

by induction [Al9 A2, , Ant]nt to be a circulant of type (nl9 n2, , %>if each of Alf A2f •• ,AH is a circulant of type (nl9n39 •• , % ^ 1 ) . For1 g i^ k let Ht = (1, Si9 SI , S?*-1) and A - [1, S*«-\ S<T2, , SJ...Henceforth we shall always let Gx, G2, , Gn denote the elements of Gin the order implied by the vector equality

(1) (Gl9 G2,...,Gn) = H1xH2x-.-xHk.

Let y(G1),y{G2), * ,y(Gn) be commuting indeterminants and define;the matrix 7 b y 7 = (viGiGj1)), 1 ^ i, j £ n. Then it can be provedby induction on k that Dλx D2x x Dk = (GiGj1), 1 Hki, 3 Sn, andhence that F i s a circulant of type (nun2, •• ,wΛ). Since any circulantof type (%!, n2, , wfc) is determined by its first row, it follows thatany circulant of type (nl9 n%9 , nk) may be obtained by assigningparticular values to the indeterminants y(G^), % ,y(Gn) in Y.

LEMMA 1. Circulants of type (nlfn29 *"9nk) with coefficients in afield K form a commutative matrix algebra containing the inverse ofeach of its invertible elements. For fixed m, all matrices X = (Xίty),1 ^ i9 j ^ m9 in which each Xi>5 is a circulant of type (nl9 n29 , nk}with coefficients in Ky form a matrix algebra containing the inverseof each of its invertible elements.

Proof. Let W= (wiG.Gj1)), 1 ^ i,j S m. Then W+ Y and aW

f o r aeK a r e c l e a r l y c i r c u l a n t s of t y p e (nl9n29

m

f n k ) . T h e {%, j) e l e -

m e n t of WY is

But this is the (i, j) element of YW. Hence WY = YW. Define

Then a straightforward calculation shows that z{GiGjτ) = z(GpG^) ifG^Gj1 = GpG^1. Hence the variables ^(G^G^1), 1 S i,j £n, are unambi-guously defined, so that WΎ is a circulant of type (nl9 n2f , nk). Thisproves the first half of the first assertion of the lemma. The rest ofthe first assertion follows from the fact that the inverse of a matrixis a polynomial in the matrix. The other assertion of the lemma isnow clear.

We let Br and i?* denote, respectively, the transpose and thecomplex conjugate transpose of the matrix B. The diagonal matrix

NORMAL MATRICES AND THE NORMAL BASIS IN ABELIAN NUMBER FIELDS 1117

whose diagonal entries are λlf λ2, , λn is denoted by diag (X19 λ2, ,\n). The zero and identity matrices with s rows and columns aredenoted by 0s and I s, respectively, and for i = 1, 2, , k, the compan-ion matrix of the polynomial x%i — 1 is denoted by F{ = [0, 1, 0, , 0]ni.

Let ζu be a primitive root of unity of order nu for 1 u S k. SetQu = (f Jr1^'-1*), 1 i, j ^ nu, and set Ω = Ωλ x Ω2 x x flfc. DefineΓw = n~ll2Ωu and Γ = n~1/2Ω. It can be shown by direct computationthat Tu is a unitary matrix. Hence, using the basic properties (X x Y)(ZxW)^XZx YW and ( I x Γ)* = X* x Γ* of the Kronecker prod-uct, it follows immediately that T is a unitary matrix.

LEMMA 2. If A is a circulant of type (nlfn2, •• 9nk) with firstrow a — (alf a2, , an) and complex coefficients, then T*AT = diag(εi> £a, , s») where the vector ε = (εx, ε2, , εJ is linked to the vectora by ε' = £?α\

Proof. The proof is by induction on k. For A; = 1 it is well known(and straightforward to check) that ATX = 2\diag (εx, ε2, •• ,e ί l l). Sup-pose the result known for k — 1. If

A = [Au A2y , Ant]nifc = ΣΛi x Ft1

a n d i f w e s e t d = ^ ^ 2 ••• w A _ i a n d d e f i n e ( 7 ( i - D ( i + i , f y « - i ) Λ + 2 > •••» ^ ) b y

/p\ i^! X X ίjA._ 1((X ( ί_ 1 ) d + 1, (X ( ί _ 1 ) d + 2 , * ' , did)

then, by the induction assumption,

\±ιX X J-k-i) ^ i i V ^ i x • • • X 1 k-i)

= C H R P * ( Ύ , - ^ , 7 , „ , , „ • • • 7 ^ 1 <Ξ ί < tlr. .

Then

— ^ - l U V •*• 1 X X •* Jfc-ij ΆiV -t l X * X 1 k-l)i •>ς-\±kJ?kJ-ki )

= Σ({diag (7 ( ί-i,β +i, 7(ί-i)ei+2, ', 7W)}

x {diag (1, fj-1, fl(i-1}, , ξ jί**-1*"-1*)})

Thus T'*AΓ is diagonal. If r = (b - ΐ)d + c where 1 c d and1 ^ δ <: wfc, then the (r, r) diagonal element of T*AT is

1118 R. C. THOMPSON

Setting ε = (εl9 ε2, , εn) and γ = (yl9 γ2, , γn), equations (3) are thesame as the matrix equation s' = (Id x Ωk)y' and equations (2) are thesame as ((Ω1 x x Ωk-τ) x /»s)α' = 7'. Combining these two facts, weobtain εf = Ωaf, as required.

3Φ The uαiqueness of the normal basis* If βGl, , βGn is anothernormal basis of F then (βG\ , β*»)' = (au)(a*ι, , α**)' so that C^'^"1)= (α^Xα^V1), lî,3^n, where ( β ^ ί 1 ) and ( α W 1 ) are both circulantsof type {nlyn2, •••,%) and (α^y) is a unimodular matrix of rational in-tegers. By Lemma 1, (aitj) = {βP&Jôfi&J1)-1 is also a circulant of type(nl9n2, * ,nk). Conversely, if βlf •••,/?„ is an integral basis such that(A, * *, βnϊ — {î,ό){a&ι, '• ,aGn)f where (<xί(i) is a unimodular circulantof rational integers of type (nl9n2,

m ,nk)9 then {βΘJι)~ (a>ij)(ocθiσϊι) so

that, by Lemma 1, {βQJι) is also a circulant. Then, in (βΘJι)> the ele-ments in the first column are a permutation on those in the first row-Hence β19 , βn is a permutation of a normal basis. Following [4], wecall a circulant trivial if it has but a single nonzero entry in each row.Thus βίf , βn is necessarily a permutation of aGl, , a?» or of — aQι,•••, — aGn precisely when all unimodular circulants of rational integersof type (nl9n2, * ,nk) are trivial.

If G has a cyclic direct factor of order other than 2, 3, 4, or 6,we may choose the notation so that (SO is this cyclic direct factor. By[4] there exists a nontrivial unimodular circulant B of rational integersof type (n^. Then B x /«2...»fc is a nontrivial unimodular integral cir-culant of type (nltn2, •• ,wΛ) and so the normal basis is not unique.Hence only the following two cases remain to be considered:

(i) each ni — 4 or 2;(ii) each nt — 3 or 2; 1 ^ i 5Ξ k.In either case (i) or case (ii) let A be a unimodular circulant of

rational integers of type (nlfn2, " ,nk). Then, by Lemma 2, the de-terminant of A is exe2 εn where each e* is an integer and hence aunit in the field K generated by ξlf , ξk. K is generated by theroot of unity whose order is the least common multiple of nl9n29 •••,nk. Since this least common multiple is 2, 3, 4, or 6, by the funda-mental theorem on units K contains no units of infinite order and henceeach ε4 is a root of unity. By Lemma 2,

(4) TV = n-ll2ε' .

Since the first row T consists of ones only, ε± is rational. In (4) wereplace, if necessary, each a{ with — a{ and each ε{ with — ε{ to ensurethat ex = 1. Since T is unitary,

(5) a' = n~ll2T*εr - n"1fl*ε f .

Let Ω = (Tij), 1 S i, 3 ^ w. Then, using (5), the triangle inequality,and the fact that each | riΛ | and each | eά | is one, we find that

(6) I αq = βj = 1, the condition for equality in thetriangle inequality forces r i i gsy = 1 for each j so that εy == ritff for j =1, 2, , w. Then, for i Φ q,

n

ncii = Σ r ^ r ^ = 0

since the columns of Ω are pairwise orthogonal. Thus, in A, there isbut a single nonzero entry in each row.

THEOREM 1. The normal basis for the integers of F is unique (upto permutation and change of sign) precisely when either (i) or (ii)below is satisfied:

(i) G is the direct product of cyclic groups of order 4t and I ororder 2;

(ii) G is the direct product of cyclic groups of order 3 and/ororder 2.

Another form of this theorem is given in [1, Theorem 6],

4, Normal discriminant matrices* Let aGl, , aQn be a normalintegral basis of F and let Δ be any normal discriminant matrix.Permuting the row and columns of Δ in the same way (this preservesnormality) we may assume Δ — {βίi1) 1 ^ ί, j ^ n, where Gl9 , Gn aregiven by (1). Now Δ = (aitj)D where D = (a0****), 1 ^ i, j g n, andwhere (aitj) is a unimodular matrix of rational integers. From ΔΔ* —Δ*Δ we get (flu)DD*(aiti)' = D*(au)'(au)D. As in [4], DD* is rationalso that D*(ait3)

f{aitJ)D is left fixed by every element of G. Let

where, here and henceforth, n0 = nk+1 — 1. The effect of replacing αwith αSs in J9 may be determined by noting that

x A

A x x A

Hence, replacing α with αss in D changes Z) into PSD. ThereforeD*(αuy(αiJ)D=(PβD)*(αi,sy(αu)(PaD) so that PXα^Yiα^P^ (αuy(αu),

A x •= Ax= L.x= pm

• X

. . .

. . .

X

AX

X

0 = A x(F.D.) x

Ins , X F,

x A ) .

~ x{S.D,• •• x A

,)

•

X

• X

1120 R. C. THOMPSON

for s = 1, 2, •••,&. Following [4] we define a generalized permutationmatrix to be a permutation matrix in which the nonzero entries arepermitted to be ± 1. Then Lemma 3 below shows that (aitj) = QC whereQ is a generalized permutation matrix and C is a circulant of type(nlf n2f ••,%). Since (βlf , /9Λ)' = (au)(a?\ ••,«*»)', this implies (byremarks made in § 2) that βl9 •••, βn is a generalized permutationof a normal basis.

THEOREM 2. Lβ£ F be a field with a normal integral basis. Thenonly generalized permutations of a normal basis can give rise to nor-mal discriminant matrices.

THEOREM 3. If A is a unimodular matrix of rational integerssuch that AA! is a circulant of type (n19 n2, , nk), then A — CQwhere C is a unimodular circulant of rational integers of type (nl9 n2r

•• ,nk) and Q is a generalized permutation matrix.

Proof. Since each Pt is a circulant of type (nl9 n29 , nk), itfollows from Lemma 1 that PiAA!P\ = AA! for i = 1, 2, •••,&, so thatTheorem 3 follows from Lemma 3.

LEMMA 3. If A is a unimodular matrix of rational integers suchthat PiAA'P'i = AA' for i = 1, 2, , fc, then A = CQ where C andQ are as in Theorem 3.

Proof. Let Ao — A and Qo — In. We shall prove by induction oni that, for 1 ^ i ^ fc, A — AtQi where Q{ is a generalized permutationmatrix and A{ may be so partitioned that A{ = (X8tt), 1 ^ s, t ^ ni+1ni+2.•• nknk+1, where each XStt is a circulant of type (nl9n2, •• , ^ i ) . Thecase i = fc is the statement of the lemma. To avoid having to give aspecial discussion of the case i = 1 we make the following definitionsand changes in notation. Recall that n0 ~ nk+1 = 1.

A one row, one column matrix is said to be a circulant of type(nQ). A circulant of type (nlf •••,%) will now be called a circulant oftype (no,nlf •••,%). We then know that A = A0Q0 where Ao is com-posed of one row, one column blocks which are circulants of type (n0)and where Qo is a generalized permutation matrix. Our inductionassumption is that for a fixed value of i with 1 g ΐ ^ f c we have A —^•i-iQi-i where we may partition A^ — (AStt), 1 <£ s, t ^ w^+i nk+1,so that each AsΛ is a circulant of type (no,nlf •• ,wi_1), and where Q -tis a generalized permutation matrix. We shall then deduce that A =A{Qi. For notational simplicity we set / = nonλ n^u g = Uini+1 *nk, h = ^ ί + 1 ^ i + 2 nk+l9 m = %^2 nim

Now AA' = A^AU so that from P,AA!P\ = AA! we deduce thati i = In, where M{ = A^PiA^. Since Af< is a matrix of rational

integers it follows that Mt is a generalized permutation matrix. SincePi and Ai-X may, after partitioning, be viewed as matrices with g rowsand columns in elements which are circulants of type (nQ, nx , %_i),it follows from Lemma 1 that ikf; is also a matrix with g rows andcolumns in elements which are circulants of type (n0, nl9 , _i). Fromthis point of view M{ must be a ' 'generalized permutation matrix" inthat it has but a single nonzero entry in each of its g rows and columns.Each of these nonzero entries is of course both a circulant of type(no,n19 •• ,ni-1) and a generalized permutation matrix.

We now digress for a moment to note that if M is a permutationmatrix whose coefficients lie in a ring with identity then a permutationmatrix R exists with coefficients in the same ring such that R'MR isa direct sum of one row identity matrices and/or matrices like [0,1, 0,• • ,0]f for ί > 1. This assertion is a consequence of the fact that apermutation may be decomposed into disjoint cycles.

Applying this fact to the ' 'generalized permutation matrix" Miy wefind that a permutation matrix R{ exists with g rows and columns inelements which are either 0/ or If such that R^M^li — N{ is a direct sumof r matrices of the following type:

0

0

0

E

EjΛ

0

0

. 0

0

EiΛ

0

0

0

•

0

... o

... o

... oif βj > 1, and Eό = (Ejtl) if eό = 1. Here each 0 = 0/ and each EiΛ isboth a circulant of type (nOf nl9 , n^) (since Rt has circulants of thistype as ''elements") and a generalized permutation matrix. Moreover,ex + e2 + + er = g. Since Nt is similar to Pt and P?< = In, then N"*= In. This implies that each e, ^ nt. We shall prove that each e3- =%. The proof is by contradiction. Suppose for at least one j that

e3- < w<β We know that f{eλ + e2 + er) = fg = n. Hence /%r > nand so r > h. Now

P4 = [0,, If, 0,, . . . , 0 ^ x 1 ,

and PiA^ = A^M,. Let ίί s = (AaΛ, ASt2, , Aβ((r) for 1 ^ s ^ g. Thenfrom P.A,.! = A^Mi it follows that: ίZ"2 = HxMiy H3 = H2Mif , HH =Hnt-iMi] Hn.+2 = HH+1Mif Hni+a — Hni+2Mi, , fi"2ni = JBΓ2Wi_1Λf<; H{h-1)H+2

= iί(fe_1)Wί+1M; , fl"(A-i,w<+8 = iϊ(Λ_1)Wί+2M ί, , fl"Aw< = H^^Mt. Hence, if-Bi = fl"(i-D 4+i for ISJ^K then HU-1)H+q = BjMr1 for 2 ^ g ^ %.

1122

Consequently,

R. C. THOMPSON

B1Mi

Bh

B1RiNi

BιRiN\

Here each BάR{ 1 ^ j ^ h, may also be regarded as a row vector withg coordinates in elements which are cireulants of type (no,n19 •••,%_!).This is so because both B3 and R{ have circulants of this type as' 'elements''.

Let X = (Xu X2, , Xg) be a row vector in which the X{ aresquare matrices with / rows and columns. Then

1Λ, X2Elt2,

Since each 2?i>ff is a generalized permutation matrix, it follows that thefirst fe1 columns of XNi are, apart from order and possible change ofsign, just the first fex columns of X; the next fe2 columns of XN{ are,up to order and sign, just the next fe2 columns of X; and, in general,columns

(7) f(e0 + e1 + + es_λ) + l ,/(β 0 + *

f(e0 + e, + + O+ ^-0 + 2,

of XiSΓi are, apart from order and sign, just these same columns in X.Here e0 = 0. This holds for s = 1, 2, , r.

Hence, in B^Nl for 1 ^ v ^ n< — 1 and fixed j , columns (7) (fora fixed value of s) are just a generalized permutation of columns (7)in BjRi. Moreover, the elements appearing in columns (7) and row qof BjRi ΐor 2 ^ q <£ / are just a permutation of the elements in columns(7) and the first row of BάRiy since B^ is composed of blocks whichare circulants of type (no,nlf •• ,ni-1). All this means that the ele-ments in columns (7) (for a fixed value of s) and row q (for 2 ^ q ^ m)of the matrix


(8) BjR.Nl

are generalized permutations of the elements in columns (7) and thefirst row of this matrix. Hence the integers in row q (for 2 ^ q ^ m)and columns (7) of the matrix (8) are congruent (modulo 2) to a per-mutation of the integers in column (7) and the first row of (8).

In the matrix A ^ i ^ add columns f(e0 + e1 + + es-±) + 1, f(e0 +eλ+ + es_0 + 2, , f(e0 + e1 + + es) — 1 to column f(e0 + ex +• + es) for s = 1, 2, , r. The integers appearing in rows mp + 2,mp + 3, , m(p + 1) of column f(e0 + eλ+ + e) are now congruent(modulo 2) to the integer in row mp + 1 and column f(eo + e1+ +es). This holds for p = 0, 1, , h — 1, and s = 1, 2, , r. Now addrow mp + 1 to rows mp + 2, mp + 3, , m(p + 1) for p = 0,1, ,h — 1. The integer in row mp + g and column f(e1 + e2 + + es) isnow congruent to zero (modulo 2), for 2 ^ g ^ m; 0 ^ p ^ & — 1; 1 ^ sg r. Hence columns /(e x + e2 + + es) for 1 ^ s ^ r may be regardedas lying in the same vector space of dimension h over the field of twoelements. Since r > h, these vectors are dependent. Consequently thedeterminant of Ai^1Ri is congruent to zero (modulo 2). This is a con-tradiction as the determinant of Ai^ΛRi is ± 1.

Hence each e3 = n{. Let Zj be the block diagonal matrix diag(/„ EύΛi EiΛEjΛ9 , EiΛESΆ JSi.^-α). Since EiΛEiΛ Ej>H is a di-agonal block in Ef and since Eψ — /m, it follows that EiΛEit2 Eίt%i

= If. From this fact and the fact that the Ej>q are generalized per-mutation matrices we find that ZJEJZJ = [0f, If, 0f, , 0/]^. Hence,if Z = diag (Zx, Z2, , Z r), then ZAΓ.Z' = P ί β Morever, Z is a matrixwith # rows and columns in elements which are circulants of type(no,nl9 •• ,ni-1). We now have Mt = U\PiUi where Ur

i — RiZt is a

generalized permutation matrix and a matrix with # rows and columnsin elements which are circulants of type (no,nlf •• ,w»-i) Then

Bjj\pγ-χ

•BhV\

1124 R. C. THOMPSON

say. Here each BάΌ\ is a vector with g coordinates in elements whichare circulants of type (n0, nly •• ,% i _ 1 ). From the form of A« it followsthat At may be partitioned into blocks which are circulants of type(nQ,nlf ••-,*&<).

The proof is now complete.

REFERENCES

1. G. Higman, The units of group rings, Proc. London Math. Soc, 46 (1940), 231-248.2. D. Hubert, Theorie des corps de nombres algebriques, Paris, (1913), 164.3. C. C. MacDuffee, The theory of matrices, New York, (1956), 81.4. M. Newman and O. Taussky, A generalization of the normal basis in abelian algebraicnumber fields, Comm. Pure Appl. Math., 9 (1956), 85-91.5. O. Taussky, Unimodular integral circulants, Math. Z., 6 3 (1955), 286-289.6. O. Taussky, Matrices of rational integers, Bull. Amer. Math. Soc, 66 (1960), 327-345.

UNIVERSITY OF BRITISH COLUMBIA

ABSOLUTE CONTINUITY OF INFINITELYDIVISIBLE DISTRIBUTIONS

HOWARD G. TUCKER

L Introduction, and summary* A probability distribution functionF is said to be infinitely divisible if and only if for every integer nthere is a distribution function Fn whose w-fold convolution is F. If Fis infinitely divisible, its characteristic function / is necessarily of theform

•( 1) f(u) = exp \iwt + ^ ( β * " - 1 - ^ - j ) - L t ί L dG{x)) ,

where u e (— 00,00), γ is some constant, and G is a bounded, non-decreasing function. J. R. Blum and M. Rosenblatt [1] have foundnecessary and sufficient conditions that F be continuous and necessaryand sufficient conditions that F be discrete. The purpose of this noteis to add to the results of Blum and Rosenblatt by giving sufficientconditions under which an infinitely divisible probability distribution Fis absolutely continuous. These conditions are that G be discontinuous

at 0 or that +• (llx2)dGac(x) = 00, where Gac is the absolutely continuouscomponent of G. In § 2 some lemmas will be proved, and in § 3 theproof of the sufficiency of these conditions will be given. All notationused here is standard and may be found, for example, in Loeve [2].

2 Some lemmas. In this section three lemmas are proved whichwill be used in the following section.

LEMMA 1. If F and H are probability distribution functions, andif F is absolutely continuous, then the convolution of F and H, F*H,is absolutely continuous.

This lemma is well known, and the proof is omitted.

LEMMA 2. // {Fn} is a sequence of absolutely continuous distribu-tion functions, and if pn^l and ££=i Vn — 1> then Σ?=i V«F* is <*>nabsolutely continuous distribution function.

Proof. By using the Lebesgue monotone convergence theorem itis easy to verify that Σ~=1 pnfn is the density of ΣZ=i PnFn, where fn

is the density of Fn.

Received November 29, 1961.

1125

1126 HOWARD G. TUCKER

LEMMA 3. Let {Y, Xu X2, •••} be independent random variables.Assume that the X/s have the same absolutely continuous distributionF, and assume that the distribution of Y is Poisson with expectationλ. Then Z = Xτ + + Xγ has a distribution function which has asaltus e~λ at 0 and is absolutely continuous elsewhere, and has ascharacteristic function

fz(u) = expλj~ (eiu* - ϊ)dF(x) .

Proof. Let E(x) be the distribution function degenerate at 0, andlet F*n(x) denote the convolution of F with itself n times. Then it iseasy to see that the distribution function of Z, Fz(z), may be writtenas Fz(z) = e~xE(z) + Σ*=i e-λ(Xnlnl)F*n(z). By lemma 1, each F*n isabsolutely continuous and has a density f*n. We need only show thatEz(z) — e~λE(z) is absolutely continuous. If we write

Fz(z) - e'λE(z) = Σ e-\\ηn\)\Z f*n(t)dt

and apply the Lebesgue monotone convergence theorem we obtain thisconclusion.

3 The theorem* If G is a bounded nondecreasing function usedin (1), then we may write G(x) = Gs(x) + Gae(x), where Gs is a singularnondecreasing function and Gac(x) is an absolutely continuous non-decreasing function.

THEOREM. Let F be an infinitely divisible distribution functionwith characteristic function (1). Then F is absolutely continuous ifat least one of the following two conditions is satisfied:

(i) G is not continuous at 0, or

(ii) Γ {llx*)dGac{x) = cχ>.J-oo

Proof. If condition (i) is satisfied, then by Lemma 1 it easily fol-lows that F is absolutely continuous, since in that case F is a convo-lution of a normal distribution with another infinitely divisible distri-bution. We now prove that condition (ii) is sufficient. By Lemma 1 itis sufficient to prove that the distribution function Fo whose charac-teristic function is

is absolutely continuous. Let εn > εn+1 > 0 for each n be such thaten —> 0 as n —• co and such that

ABSOLUTE CONTINUITY OF INFINITELY DIVISIBLE DISTRIBUTIONS 1127

λw = ^ ((1 + x*)lx*)dGac(x) > 0 ,

where

Sn = ( - ε ^ , -en] U [en, en_λ) , n = 1, 2, . . . ,

and where ε0 = co. Let Un be a random variable with characteristicfunction

/ Q \ n / v f / ^ , , 1 iUX M

and let

((1(-<*>.x]Γ\8n

One easily sees that λw < oo and that £Γw(α?) is an absolutely continuousdistribution function of a bounded random variable. For each positiveinteger n we may write, by Lemma 3, that

Un = XnΛ + Xn>2 + . . . + Xn,zn - \ (llx)dGac(x)

where Zn is a random variable with Poisson distribution with expecta-tion λn, where {XnΛ, Xn,2, •} have the common absolutely continuousdistribution function Hn(x), and where {Zn, XnΛ9 Xn>2 •} are independent.If we assume that

{{Zn, XnΛ, Xn.if .••},% = 1,2, -..}

are all independent, then the distribution function of

Uo = Σ Un = £ ( Σ Xnj -\ (llχ)dGac(x))n=l n = l \j=l JSn /

is equal to Fo. Now let us define a sequence of events {Cn} by

Cr = [Zx Φϋ\, C, = [Z, = 0][Zt Φ 0] ,

and, in general,

These events are easily seen to be disjoint. If we define

( 4 ) Co = ( ϋ C.Y= ή [Z. = 0] ,

then Ω = Un=i Cw, where β is the sure event. The distribution func-tion of Z70 is

1128 HOWARD G. TUCKER

FUa{u) = Σ P([U0 £ u] I Cn)P(CJ + P([Ut ^ u]C0) .

By (4) and by hypothesis, we obtain

P([U0 ^ u]C0) rg P(C0) = l i m e x p ί - ("8n + \~(llx*)dGae(x)\ = 0 .n^co I J_co Jsn )

Also, P([ Uo S u] I Cn) is the distribution function of XnΛ + Wn9 whereXnΛ and Wn are independent random variables. Since the distributionfunction of XnΛ is absolutely continuous, it follows by Lemma 1 thatP([ Uo ^ u] I Cn) is absolutely continuous for each n. Lemma 2 thenimplies that FUQ(U) is absolutely continuous, which concludes the proofof the theorem.

The condition given in this theorem is not necessary, as is shownin the following example. Let γ = 0 in (1), and let α, β be real numberswhich satisfy β > 1,1 > a > β/2. For j = 1, 2, , let us denote

Xj = j~* and pj = i " β .

Let G be a pure jump function with jumps at x3- and — x3- of size ρ3

for every i . (The total variation of G is 2 ^ f t < °°.) In this case weobtain

f(u) - exp 2 Σ (cos J L -

We shall show that there is a constant iΓ such that for all | u \ π,the inequality

(5) 0

is true. This is equivalent to showing that

(6) Σ n** + 1 sin2-L -L > K\^ 2tlα

Let us consider, for each fixed | u\ ^ π the integer N defined by

where the square brackets have their usual meaning. It is easy toverify that 0 < | u |/2iVα < π/2, and thus we may write

\u\

2Γ(2' ΐ t'Y/T

ABSOLUTE CONTINUITY OF INFINITELY DIVISIBLE DISTRIBUTIONS 1129

where K does not depend on u. This inequality implies that inequality (6)is true, thus implying (5). Inequality (5) implies that f(u) e Lx(— oo, + °°),which in turn implies that f(u) is the characteristic function of anabsolutely continuous distribution. (See Theorem 3.2.2 on page 40 in[3].)

I wish to acknowledge several helpful suggestions by my colleague,Professor H. D. Brunk. The example just outlined was suggested bythe referee to whom I wish to express my appreciation.

REFERENCES

1. J. R. Blum and M. Rosenblatt, On the structure of infinitely divisible distributions,Pacific J. Math., 9 (1959), 1-7.2. M. Loeve, Probability Theory, D. Van Nostrand, Princeton, 1960 (Second Edition).3. Eugene Lukacs, Characteristic Functions, Hafner, New York, 1960.

UNIVERSITY OF CALIFORNIA, RIVERSIDE

COMPLETELY DISTRIBUTIVE LATTICE-ORDEREDGROUPS

ELLIOT CARL WEINBERG

Loosely speaking, a lattice is called distributive if the order ofperforming the operations of finite suprema and infima may be inter-changed. A lattice is called completely distributive if the order ofperforming the operations of infinite suprema and infima may be inter-changed. The purpose of this note is to relate the property of completedistributivity in^ Z-groups to other Z-group properties. We shall prove,for example, that an Z-group G which has an atomistic lattice of polarsis completely distributive. The most interesting results are obtainedfor Archimedean Z-groups. In this case the two above-mentioned pro-perties are equivalent to each other and to the existence of certain 'nicerepresentations of G as subdirect unions of simply ordered groups. Inthe last section examples are given to distinguish these properties fornon-Archimedean groups.

l Preliminaries* We shall follow the notation and terminology ofChapter XIV of [1], to which the reader is referred for general back-ground concerning Z-groups.

1.1. DEFINITION. Let i f = {C{: iel} be a family of simply orderedgroups. The complete direct union of ^ is the i-group of all functionsα : J — > U ^ such that α, = a(i)ed for all iel with the operationsdefined by (aV&)* = α<Vb{ and (α + b)t = a{ + &< for all iel. Thediscrete direct union of ^ is the Z-subgroup of the complete directunion which consists of those functions which are zero at all but afinite number of points of I. The Z-group if is a subdirect union of^ if it is an Z-subgroup of the complete direct union for which theprojection map Pi of H into the factor group C{ maps H onto C{ foreach iel. The subdirect union H of the family <& is called regular ifthe projection map Pi is a complete lattice homomorphism for each iel;i.e., Pί(VjejQj) — VjejPiiOj) f° r each iel. (More generally, a sublatticeL of a lattice M is called a regular sublattice if the injection map ofL into M preserves infinite suprema and infima.) A complete subdirectunion of ^ is an Z-subgroup of the complete direct union which con-tains the discrete direct union. An Z-group is called representable if itis isomorphic to a subdirect union of simply ordered groups.

Recall that a polar of an Z-group G is a subset of G which consistsof the elements disjoint from each element of some subset of G.

Received October 18, 1961.

1131

1132 ELLIOT CARL WEINBERG

Partially ordered by inclusion the set of polars of an Z-group forms acomplete Boolean algebra. The role of polars in the study of Z-group&is indicated by the following proposition, a proof of which may be foundin [3].

1.2. PROPOSITION. Let G be an Z-group.(a) If J is an Z-ideal of G, then G/J is simply ordered if and only

if J contains each element of some prime ideal in the Boolean algebraof all polars of G.

(b) Each polar of G is an Z-ideal if and only if G is representable.(c) G is isomorphic to a complete subdirect union of simply ordered

groups if and only if each polar of G contains a minimal direct factorof G.

1.3. PROPOSITION. Let J be an Z-ideal of the Z-group G. In orderthat the canonical homomorphism φ:G-^ G\J be a complete latticehomomorphism it is necessary and sufficient that J be closed (i.e., if{ji'. iel} c J and \fiei3i exists in G, then

Proof. Assume that J i s a closed Z-ideal. Let g = Vίer#;(#> 9ίe G)Then Aίei(9 — 9i) = 0. Suppose that heG+ satisfies 0 ^ φ(h) ^ φ(g — &)•for each iel. Then, for each iel, there exists h{ e J Π G+ such thath ^ g — Qι + h^ from which we can calculate 0 V (h — h) ^ Qi — ΰ and

0 - Aieiiff ~ ft) = Aieilih -hi)V0]=h+ Λ<ei(-Λ< V - h), SO h =V ei(^Λ^ )- Since J is convex, hAh^J', since J is closed, λe J.Hence φ(h) = 0 and AieMo - Qi) = (ff) - VieMΰd = 0. The converseis obvious.

1.4. COROLLARY. / / α poZαr / o/ cm l-group G is an l-ideal,.then the homorphism φ:G-+ G/J is complete.

Proof. Let J be the set of all elements of G disjoint from eachelement of the set H. Suppose that { : ίel} d J and Vίe/iί exists inG. It suffices to assume that j\ ^ 0 for each iel. Let heH. SinceΛ Λ | Λ | = 0 for each ΐ, (Vίe/Λ) A I A I = V<6/(i A 1 h |) = 0. Thus

J.

1.5. DEFINITION. An Z-group G is completely distributive if Λ eiV;ej#u = VφejiAiei9i<P(i) whenever {gi3: i e I, j e J} is a subset of Gfor which all of the indicated suprema and infima exist.

1.6. PROPOSITION. Let G be an Z-group. The following are equiva-

lent.

COMPLETELY DISTRIBUTIVE LATTICE-ORDERED GROUPS 1133

(a) G is completely distributive.(b) If {gi:j:ieI,jeJ}(zG+ and 0 < g = \f serfu f o r e a c h ίeI>

then there exists φeJ1 such that AieiQiφn) = 0 is false.(c) If 0<geG, then there exists g* > 0 such that g = AieiQi

implies g* < g{ for some ie I.

Proof. The equivalence of (a) with (b) is discussed in [5], whilethe equivalence of (b) with (c) is clear.

1.7. PROPOSITION. Let G be an i-subgroup of H such that eachpositive element of H is the supremum of some family of elements ofG. If G is completely distributive, then H is completely distributive.

Proof. Suppose that 0 < h e H. There exists g e G such that0 < g ^ h. Let g* be the element whose existence is guaranteed by1.6(c). Suppose that h = Vίeih where {hi'.iel} c H+. For each ielthere exists a family {gij:jeJ}czG+ such that ht = VίejΛi Then

g = gAh = gA(V<€Λ) = gA(V*er./6j&y) = V*.i(ff Afty).

There exists a pair ielel,jej such that g*^gi5. Hence g*^hi9

so iJ is completely distributive.

2. The main result. Let G be an Z-group. Consider the followingproperties.

(A) G is isomorphic to a complete subdirect union of simply orderedgroups.

(B) The Boolean algebra P(G) of polars of G is atomistic.(C) G is isomorphic to a regular subdirect union of simply ordered

groups.(D) G is completely distributive.

2.2. THEOREM. / / G is representable, then (A) =φ (B) =Φ (C) =#> (D).7/ G is Archimedean, then the four properties are equivalent. Forarbitritary l-groups, (B) =φ (D).

Proof.(A) implies (B). This is an immediate consequence of 1.2(c).(B) implies (C). Observe that (C) is equivalent to the requirement

that there exist a family of ί-ideals L; such that each GIL{ is simplyordered, each homomorphism G —> G/L< is complete, and Π Λ — {0}. Letα be a strictly positive element of G. There exists a maximal polar Lα

containing the set of all elements disjoint from a but not containinga. Since G is representable, each La is an {-ideal. By 1.3 and 1.2(a),

the family {La: 0 < a e G} has the requisite properties.(C) implies (D). Simply ordered groups, the complete direct union

of a family of simply ordered groups, and regular Z-subgroups of suchgroups are completely distributive. The Z-group G belongs to the lastcategory.

If G is Archimedean, then (D) implies (A). Let G^ denote thecompletion of G [1, p. 229]; i.e., CL is a complete Z-group which con-tains G as an Z-subgroup in such a way that each element of CL is thesupremum of the set of all elements of G which it contains. By 1.7,Goo is completely distributive. We shall first show that P(Goo) is atom-istic. If P is a nonempty polar of G , then in contains a strictlypositive element/. Consider the family {{fio fujiie 1} of all pairs ofelements of Gt such that / = /<oγ' fiτ for each iel. Since CL is com-pletely distributive and complete, there exist φe2Σ and heG^ such that0 < h = Aieifiφw We claim that h, the smallest polar of G^ whichcontains h, is an atom which is contained in P. Indeed, since h ^ /,h c P. Now, recalling from [1, p. 233] that every closed Z-deal of acomplete Z-group is a direct factor, we see that if K and K' arecomplementary polars of GL, then G^ is the direct union of K and Kr.Thus there exist keK and k'e K' such that / = k + k' = fc V&', fromwhich it follows that h ^ k or h <£ kf. Hence h c K or h c K'. Sinceh is contained in one element of each complementary pair of P(Goo), itis an atom.

Now observe that the map P—*PΓ\G is an isomorphism of P(Goo)onto P(G). Thus to complete the proof it suffices to show that, foreach atom A of P(G«,), A Π G is a direct factor of G. Let A' denotethe complement in P(G00) of A. Let feG+. There exist unique ele-ments a e A and a9 e A' such that / = a + α'; moreover, α and α' belongto Gi. If α' = 0, then f = aeAΓiG, so suppose a' > 0. There existsxeG such that 0 < x g α'. Since G is Archimedean, there exists apositive integer n such the nx j£ α'. Since A is an atom of P{G00)1 itis simply ordered, so a' ^ nx; moreover, x ^ a' implies that x e A!. Wecan calculate f Anχ — (a V a') A nχ — (a A ^ ) V (α ' A ^^) — 0 V a ' — α'»so αf e G and a = f — a' eG. This completes the proof.

(B) implies (D). Let / be the smallest polar of G containing thestrictly positive element /. If a is an element of an atom A of P(G)contained in / such that 0 < α, then 0 < a J\fe A. Since A is a simplyordered Z-subgroup of G, it is completely distributive, so there existsa* e A such that a A/ = Viei^i £ A+, implies 0 < α* ^ a{ for some ΐ e /.Suppose that / = Vieifi,fieG+. Than aAf = V<e/(αΛ/<), and αΛΛG A for each ΐ 6 /, so there exists iel such that 0 < α* ^ a A/; ^ /<•In other words, G is completely distributive.

2.2. REMARKS. The observant reader will have noticed that the

proof that (D) implies (A) may be modified to prove that a completelydistributive i-group G has an atomistic lattice of polars provided thatG may be embedded as an i-subgroup of a group H in which eachpolar is a direct factor and each positive element is the supremum ofa set of elements of G. There are i-groups for which such embeddingscannot be found (see. 3.4.)

Some of the implications of 2.1 are suggested by earlier lattice-theoretic propositions. That (B) and (D) are related sounds like Tarski'stheorem [4] that a Boolean algebra is completely distributive if andonly if it is atomistic. That (C) and (D) are related sounds like atheorem of Raney [2] which says that a complete lattice is completelydistributive if and only if it is isomorphic to a regular sublattice of acomplete direct union of chains.

3. EXAMPLES. We will exhibit the following examples.

3.1. A representable i-group which satisfies (B) but not (A).

3.2. A representable i-group which satisfies (C) but not (B).

3.3. A nonrepresentable i-group which satisfies (D) but not (B).

3.4. A completely distributive i-group whose completion by cuts isnot completely distributive.

Unfortunately, we do not know if there exists a representable l-group which satisfies (D) but not (C).

3.1. Consider the lexicographic product J'© J2 of the i-group J ofintegers and the direct union J2 of two copies of J. This i-group hasprecisely two proper nonzero polars, but it has no direct factors. Thedetails have been discussed in [3],

3.2. Let L = ©iejyJi be the lexicographic sum of a countablefamily of copies of the i-group J of integers indexed by the set N ofpositive integers. Let Q denote the set of rational numbers of theform p/2fc for p = 0, 1, , 2k - 1 and keN.

Let

HPtk = {xeQ: p\V ^ x < (p + l)/2*} .

Denote by G the set of all elements g of the i-group LQ of all functionsfrom Q into L such that there exists k = k(g) e N satisfying the twoconditions:

(i) if k < m, then g(x)(m) = 0 for all xe Q, and(ii) if m <: k, then g(x)(m) = g(y)(m) for all x,ye Hp,m. We shall

show that G is an ϊ-subgroup of LQ which has the desired properties.It is clear that g,heG, implies g-h e G, so G is a partially ordered

group. Let geG, and denote by g V 0 the supremum of g and 0 in ZΛSince L is simply ordered, for each xeQ, either (g V 0)(x) = 0 or (g V 0)(x) = 0(a>). Certainly, (gV0)(x)(j) = 0 for all i > &(#) and for sΆxeQ.Now let a?, ye Hpj for some j" &(#). Observe that XG Hq>m for m ^ jif and only if yeHq,m. If (g\/0)(x)(j) φ (g\/O)(y)(j), then we mayassume, since g(x)(j) = g{y)U), that (#V0)(#) = 0 while 0 < 0(2/) =(0 V0)(l/) If m is ^he first integer such that g(y)(m) Φ 0, then m ^ j .Let g be an integer such that yeHQtm. Then xeHq>m, 0 < #(αθ(m) =g(y)(m), and #(α?)(m') = 0 for all m' < m. Hence g(x) > 0. With thiscontradiction we have completed the proof that g V 0 e G. Hence G isan J-subgroup of ZΛ

For each x e Q, let M, = {# e G: g(x) = 0}. The reader can easilyverify that each Mx is an Z-ideal of G, that Π*eβ^ = {0}9 and thatG/Afβ, being isomorphic to L, is simply ordered. We shall prove thateach Mx is closed. Suppose, on the contrary, that there exists a set{g{: ίe 1} c Mx such that Vie/ i = 9 while #(#) = a > 0. Let α denotethe function in LQ with constant value α. Let hi — (a — g{) V 0. ThenΛieih = (α — Vieiθi) yθe Mx, while (α?) = α for each ΐ e / . We canshow, however that {h{:iel} has a positive lower bound not in M9, acontradiction which will complete the argument that G satisfies property(C). Indeed, let j be the first integer such that a(j) > 0. (Supposethat x = p/2\) Then ^(p/2fc)(i) = a(j) > 0 for each ie I. If k ^ i,then hi(y)(j) = α(j) for all yeH^-tj c i ί ^ . In this case each / isbounded below by the function / defined by

(i) for yeHp2j-^p f(y)(j) - a(j), f(y)(j + 1) = α(i + 1) - 1, andf(y)(m) = 0 otherwise; and

(ii) for 2/ 0 fl'^-*ii,/(l/) = 0.If i < k, then let 6 e L be given by 6(&) = 1 and b(m) = 0 for m Φk.Then 6 < α. Letting /» = fc< A ^ we see that 0 < /< and /ί(p/2fc) = &for each ίe I. Now the previous case applies to the new set of func-tions {fi'.ie I}.

It is easy to verify that P(G) is not atomistic. For geG, let Z(g)= {xeQ: g(x) = 0}. Since | g | A I h \ - 0 if and only if Z(g)\JZ(h) = Qr

while, for each ge G, we can find heG such that | g \ A I h \ = 0 and^(^) = Q — -2ΓW, we conclude that the smallest polar containing anelement g consists of all elements h such that Z(g) = Z(h). Hence thefamily {HPtk: ke N; p = 0, 1, , 2fc — 1}, partially ordered by inclusion,is isomorphic to a coinitial subset of the partially ordered set of nonzeropolars. Since the first family has no minimal elements, P(G) has noatoms.

3.3. Consider the i-group C of one-to-one order preserving maps


of the closed unit interval onto itself ((/ + g)(x) = / (g(x)) and / ^ g ifand only if f(x) ^ g(x) for all x). That C is completely distributivefollows from the fact that C is a regular sublattice of the lattice of allfunctions from the unit interval into itself. That P(C) is not atomisticfollows from the fact that each strictly positive element of C containstwo disjoint strictly positive elements.

3.4. We again call upon the ί-group of example 3.1. Let J2 denotethe lattice obtained by adjoining a largest element (oo? oo) to J2. It iseasily verified that the completion of J®J2 is J@J2.

The conditionally complete lattice J 0 J2 is not completely distribu-tive. For each pair of integers (α, 6), let fab = (1, (α, &)). Then

Aaej Vυejfao = (1, (°°, «>)), whileVφβJlAaejfaφia) = (0, (OO , Oθ)) .

This example is interesting for another reason. The Z-group J φ / 2

is maximal in the sense that it cannot be embedded as a proper ί-subgroup of any ϊ-group H in such a way that each element of H isthe supremum of a set of elements of J 0 J2.

REFERENCES

1. G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloquium Publications, rev. ed., 2 5(1948).2. G, N. Raney, A subdirect union representation for completely distributive completelattices, Proc. Amer. Math. Soc, 4 (1953), 518-522.3. F. Sik, Uber Summen einfach geordneter Gruppen, Czech. Math. J., 8 (1958), 22-53.4. E. C. Smith Jr. and A. Tarski, Higher degrees of completeness and distributivity inBoolean algebreas, Trans. Amer. Math. Soc, 8 4 (1957), 230-257.5. E. C. Weinberg, Higher degrees of distributivity in lattices of continuous functions,.Trans. Amer. Math. Soc, 104 (1982), 334-346.

UUIVERSITY OF ILLINOIS

A NOTE ON THE PRIMES IN A BANACH ALGEBRAOF MEASURES

JAMES WELLS

l Introduction. Let V denote the family of all finite complex-valued and conuntably additive set functions on the Borel subsets ofR+ = [0 oo) (hereafter called measures); L1(R+) the set of all complex-valued functions on R+ which are integrable in the sense of Lebesgue,identifying functions which are 0 almost everywhere; and A the ele-ments in V which are absolutely continuous with respect to Lebesguemeasure. For each μ e V there exists an / e L1(R+) such that

(1.1) μ(E) = \ f(x)dx

for each Borel subset E of R+. And, conversely, if fe L\R+) the setfunction μ defined by (1.1) is a measure.

We introduce a norm into V by the formula

(1.2) \\μ\\ = supΣ\μ(Ei)\ (βeV),

the supremum being taken over all finite partitions of R+ into pairwisedisjoint Borel sets Ei9 It is well known ([6], p. 142 or [7]) that Vbecomes a commutative Banach algebra under the convolution operation

(1.3) v(E) = [°μ(E - x)d\{x) (μ, λ e 7 ) ,Jo

where E is any Borel subset of R+; in symbols

(1.4) v = μ * λ .

The Laplace-Stieltjes transform of μ e V will be denoted by μ:

(1.5) μ(z) = [~e-*dμ(x) (Re(z) 0) .Jo

The relation (1.4) is equivalent to

(1.6) ί){z) = μ(z)X(z) (Re(z) ϊ> 0) .

The identity in V is the measure u such that u(E) = 1 if 0 e Eand 0 otherwise. A measure μ is invertible provided there exists ameasure μ"1 such that μ * μ-1 = u; and the measure λ is a divisor ofthe measure μ, in symbols λ | μ, provided there exists a measure v suchthat μ = λ * v. It follows from basic properties of the Laplace-Stieltjes

Received December 4, 1961.

1139

1140 JAMES WELLS

transform that V is an integral domain and a semi-simple Banach alge-bra (see for example [6], p. 149).

The central problem under consideration here is that of determiningthe prime measures, that is, those noninvertible measures μ such that

(i) μ = λ * v always implies that one of the measures λ, v isinvertible.

It is clear that every prime measure μ satisfies the condition(ii) V*μ c F * λ implies that either λ is invertible or μ\X.

And (i) follows from (ii) since V is an integral domain. Here V*μdenotes the ideal {v*μ\v e V}.

We give a partial solution by showing that all measures of theform

(1.7) μa = — - — u-η (Re(a) > 0) ,1 + a

where drj(x) = e~*dx, are primes. Stated in terms of the ideal structureof V, the result is that the maximal ideals ma = {μ \ μ(a) = 0}, Re(a) > 0,are principal.

A related problem is the following: Given a fixed measure μ, forwhat measures λ is it true that X\μ! Climaxing a sequence of paperson this problem, notably [4] and [8], Fuchs [3] proved that X\μ if andonly if the HausdorίF method of summability [H, μ] includes the method[H, λ]. In this paper we make use of recent results on the representa-tion of linear transformations by convolution to give a simple, andapparently unnoticed, alternative formulation in terms of the range ofa convolution transform.1

THEOREM 1. Every measure μa, Re(ά) > 0, is a prime; and ifthere exists a prime μ essentially different from μa9 Re(a) > 0 (twoprimes are essentially different if one cannot be obtained from theother by convolution with an invertible measure) then either μ(z) hasa root with real part 0 or the hull of the ideal V * μ consists only ofmaximal ideals in V which contain A.

THEOREM 2. Let Tμ, μ e V, be the linear operator from L\R+)into L1(R+) defined by

(1.8) Tμf(t) = / * μ(t) - (V(« ~ x)dμ(x)Jo

for f € L\R+). Let R^ denote the range of Γμ. Then the measure Xis a divisor of the measure μ if and only if Rμ c Rλ.

1 The author is indebted to the referee for his helpful suggestions.

A NOTE ON THE PRIMES IN A BANACH ALGEBRA OF MEASURES 1141

2 Proofs of the Theorems*Proof of Theorem 1. The positive result of this theorem depends

on the obvious fact (see condition (ii)) that if the maximal ideal m inV is principal and μ is a generator, that is m = V * μ, then μ is aprime.

Fix Re(a) ^ 0 and set h{μ) = μ(a). It follows from (1.5) and (1.6)that h defines a multiplicative linear functional on V. Hence ma ={μ e VI β(a) — 0} is a maximal ideal in V. That V * μa c mα followsfrom (1.4), (1.6) and the fact that μa(z) = (1 + α)"1 — (1 + z)~x vanishesat a.

The reverse inclusion requires that if μ e ma, then μ — v*μa forsome v G V. To this end we use a device suggested by [9] and define

•(2.1)

where

(2.2) dθa =

The equality of the two integrals is a consequence of μ(a) ~ 0. In caseσ = Re(a) > 0, an application of the Fubini theorem using the secondintegral in (2.2) yields

[°\f(x) \dx=\~\ [°e-a{t-χ)dμ(t) dx ^ ( " ( V ^ - Ή | μ(t) \ dxJO Jo I Ja; Jo Ja;

= [~\'er'«->dxd I μ(t) I = i- ("[1 - e~σt]d \ μ(t) \Jo Jo σ Jo

This proves / e L\R+) so that, in view of (1.1), θa e A when Re(a) > 0.It remains to verify that

μ = v*μa = (l + a)[μ + (1 + a)θa] * [(1 + a)-λu - rj\

= (1 + α)[(l + α)~7* - ^ * ) ? + ^ α ~ ( l + α)^α * rj\ .

But integration by parts yields the relation

e~a{y-χ)dμ(y)dx = (1 + αJ^ΓίV^-^d/id/) +

which, together with the fact that d(φ * y)(x) = (f * y)(x)dx wheneverdφ{x) =f(x)dx,feL1(R+) and γe V, shows that (1 + ά)θa*y = —μ*y) + θa.This establishes the result.

If jM is a prime essentially different from μa, Re(a) > 0, and μ(z)has no roots with real part 0, then μ(z) has no roots. To see this notethat μ(a) = 0 for Re(a) > 0 implies that 7 * / ι c F * μα = mα. Hence

1142 JAMES WELLS

μ = v * μa for some v e V which, because of condition (ii), forces v tobe invertible; so μ is not essentially different from μa. Thus V*μ isnot contained in mα for any a, Re(a) ^ 0. Phillips ([6], p. 148 or [7]>has shown that in the space Δ of maximal ideals in V, Δλ = {ma | Re(a) ^ 0}is precisely those maximal ideals which omit an element of A so thatΔ2 — Δ — Δλ consists of all those maximal ideals which contain A. It isclear, then, that the hull of V* μ, i.e., all maximal ideals which containit, must be a subset of Δ2.

Proof of Theorem 2. First suppose that λ | μ. Then μ — v * λ forsome v e V and, therefore,

L\R+) *μ = L1(R+) * v * λ c L 1 ^ ) * λ ,

i.e., i?μ c Rκ.For the converse we note that the inclusion R^ c Rλ implies that

for each / e Lλ{R+) there exists a g e L1(R+) such that

(2.1) f * μ = g*χ.

But the fact that V is an integral domain insures the uniqueness of g.Hence the relation (2.1) defines a mapping T: f—*g which is linear,commutes with convolution in the sense that T ( / * 7 ) = Γ ( / ) * 7 for/ 6 L\R+), 7 G F, and, via an application of the closed graph theorem,bounded in the norm topology of L\R+). It follows using the type ofargument given in [2], that every such mapping has the form T(f) —f *v for some measure v. Thus

(2.2) f*μ = ( /*y)*λ = /*( i ;*λ)

for every / e L\R+). A second application of the fact that V is anintegral domain yields μ = v * λ, that is λ | μ, and the theorem is proved.

3. A remark and a question. Let Re(a) > 0, Re(b) > 0. It iseasy to verify that (z + l)/(z + b) is the Laplace-Stieltjes transform ofan invertible measure. Consequently the measure defined by

(3.1) μ(z) = £ ^ f = μa(z) d + *X* + D {Re{z) > 0)z — b z + o

is a prime not essentially different from μa. The primes given by rela-tion (3.1) coincide with those given in [4]. Existence of other primesremains an open question.

Repeated application of Theorem 1 yields the relation

(3.2) V * μai * μa2 * * μan = f\ m H , n = 2, 3, *

A NOTE ON THE PRIMES IN A BANACH ALGEBRA OF MEASURES 1143

where Re{a{) > 0, i = 1, 2, 3, . On the other hand, it is known [1]that the closed ideal m = p|Γ=i H is not trivial in case ΣΓ=i 1/1 aί I < °°A natural question to ask is the following: Does there exist a measureμ such that F * μ = m?

REFERENCES

1. T. Carleman, Uber die approximation analytischer funktionen durch lineare aggregatevon vorgegeben potenzen, Arkiv for mat., astronomi och fysik, 17, 9 (1923), 1-30.2. R. E. Edwards, Representation theorems for certain functional operators, Pacific J.Math., 7 (1957), 1333-1339.3. W. H. J. Fuchs, A theorem on Hausdorff's methods of summation, Quarterly J. ofMath., 16 (1945), 64-77.4. H. L. Garabedian, Einar Hille and H. S. Wall, Formulations of the Hausdorff inclu-sion problem, Duke Math. J., 8 (1941), 193-213.5. Einar Hille and J. D. Tamarkin, Questions of relative inclusion in the domain ofHausdorff means, Proc. Nat. Acad. Sci., 19 (1933), 573-577.6. Einar Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc.Coll. Publ., XXXI, 1957.7. R. S. Phillips, Spectral theory for semi-groups of operators, Trans. Amer. Math. Soc,7 1 (1951), 393-415.8. W. W. Rogosinski, On Hausdorff's method of summability, Proc. Cambridge Phil. Soc,38 (1942), 166-192.9. L. L. Silverman and J. D. Tamarkin, On the generalization of Abel's theorem forcertain definitions of summability, Math. Z., 29 (1928), 161-170.

DECOMPOSITION AND HOMOGENEITY OFCONTINUA ON A 2-MANTFOLD

H. C. WISER

1. Introduction* Many partial results have been obtained in at-tempting to characterize homogeneous plane continua; a history of thisproblem can be found in [4]. The question arises; which of these resultshold for homogeneous proper subcontinua of a 2-manifold, and indeeddo there exist such continua which cannot be embedded in the plane?The main purpose of this paper is to extend some results for planehomogeneous continua to corresponding results for continua on a 2-manifold, with a long range aim of investigating the embedding problem.

Let Xbe a nondegenerate homogeneous plane continuum. F. B. Jones[10] has shown that X is a simple closed curve if it is aposyndetic orif it contains a noncutpoint, H. J. Cohen [7] has shown that X is asimple closed curve if it either contains a simple closed curve or isarcwise connected, and R. H. Bing [3] has shown that X is a simpleclosed curve if it contains an arc. In §4 the above results of Cohen'sand Jones' are generalized to homogeneous continua on 2-manifolds.Section 3 contains results on collections of continua which arise rathernaturally in considering the generalizations of Cohen's work.

Jones [12] has shown that if X is decomposable and is not a simpleclosed curve, at least it becomes one under a natural aposyndetic decom-position. In § 5 this result is extended to homogeneous continua on a2-manifold as well as to homogeneous continua with a multicoherencerestriction.

In extending plane results to results on arbitrary 2-manifolds, wewill use as a generalization of the Jordan curve theorem the fact thatfor any 2-manifold M there exists a positive integer k such that M isseparated by the sum of any k disjoint simple closed curves on M.

2Φ Definitions* Only separable metric spaces will be consideredhere. A connected compact metric space is called a continuum. A 2-manifold is a continuum such that each of its points lies in an openset topologically equivalent to Euclidean 2-space. A 2-rnanifold withboundary is a continuum such that each of its points lies in an open setwhose closure is topologically equivalent to a closed 2-cell.

A point set X is said to be n-homogeneous if for any n pointsxl9 %2, "*,^n of X and any n points yl9 y2, , yn of X there is a home-

Received July 5, 1961. This paper is part of a thesis submitted to the faculty of theUniversity of Utah in partial fulfillment of the requirements for the Ph. D. degree, June,1961. The author is indebted to Professor C. E. Burgess for his encouragement and advice.

1145

1146 H. C. WISER

omorphism of X onto itself that carries xx + x2 + + xn onto yx + y2

+ + Vn- For n = 1, the term homogeneous is used. A set X issaid to be nearly homogeneous if for any point a? of X and open set Dof X there exists a homeomorphism of X onto itself carrying x intoZλ A set X is locally homogeneous if for each two points x and 2/ ofX there exists a homeomorphism between two open subsets of X con-taining x and 2/ respectively such that x is mapped onto y.

A continuum X is said to be aposyndetic at the point p of X iffor any point g of I - p there is a subcontinuum Y of X and an opensubset U of X such that I - g D Γ 3 U "D p. The continuum X is saidto be aposyndetie if it is aposyndetic at each of its points.

A continuum X is said to be semί-locally connected at a point p iffor each positive number ε there exists a positive number 8 such thatX— Vs(p) is contained in a finite number of components of X— Fδ(p),(Note: In general, Vr(X) is the r-neighborhood of the set X; i.e., theset of all points x such that the distance, p{x, X), from x to X is lessthan r.) If X is semi-locally connected at each of its points, X is saidto be semi-locally connected.

A simple triod is the sum of three arcs each having a point p asan end point such that p is the common part of each two of thesethree arcs.

If G is an upper semi-continuous collection filling a continuum X,the decomposition space relative to G will be denoted by X'. Theprojection map of X onto X' relative to G will be denoted by / through-out this paper.

3 Collections of continua which fill a continuum. We will statea theorem and a corollary, from G. T. Whyburn [17, pp. 43-44], whichare needed in the proofs of some of the theorems of this section.

THEOREM W. // G is any uncountable collection of disjoint cut-tings of a connected set My then some element X of G separates inM a pair of points belonging to G* — X.1

COROLLARY W. NO continuum of convergence K of a connected setM contains an uncountable collection of disjoint cuttings of M. In-deed, if a and b are points of K, no subset of K separates a and bin M.

THEOREM 1. If G is a nondegenerute collection of disjoint continuafilling a continuum X on a 2-manifold M, Go is a countable subcol-lection of G, k is an integer such that M is separated by the sum of

For any collection G, G* denotes the sum of the sets of G.

DECOMPOSITION AND HOMOGENEITY OF CONTINUA ON A 2-MANIFOLD 1147

every k elements of G ~ Go, and D is a complementary domain of X,then the boundary of D is the sum of a finite number of continuaBu , Bm each lying in some element of G.

Proof. It follows from work of J. H. Roberts' and N. E. Steen-rod's [14, Lemma 1] that the boundary of D is the sum of a finitenumber of continua Bl9 •••, Bm. Suppose Bx intersects each continuumof an uncountable subcollectίon Gx of G. There exists an uncountablecollection Z such that each element of Z is the sum of k continua ofG1 — Go and no two distinct elements of Z intersect. It follows fromTheorem W that there is an element Q of Z such that M — Q is thesum of two mutually separated sets Hλ and H2 containing two continuag1 and g2, respectively, of Gx — Go. This involves a contradiction, sinceD does not intersect Q, and each of gλ and g2 contains a boundary pointof D. Thus, since a continuum cannot be the sum of a countablenumber (greater than one) of disjoint closed sets, Bx must be containedin some element of G.

COROLLARY 1.1. Under the hypothesis of Theorem 1, let G2 be theset of elements of G — GQ not intersecting the boundary of any com-plementary domain of X; then any k elements of G2 separate X, G2 isuncountable, and f(G2) is dense in Xr.

Proof. Suppose C is the sum of k elements of G2, M — C is thesum of two mutually separated sets H and K, and X — C c H. Allof the complementary domains of X must then lie in H, contradictingthe existence of K. Thus C must separate X. From Theorem 1, sinceX has at most a countable number of complementary domains, at mosta countable number of elements of G intersect the boundary of a com-plementary domain of X; therefore G2 is uncountable and f(G2) isdense in X'.

THEOREM 2. If G is a collection of disjoint continua filling acontinuum X in a connected space M, Go is a countable subcollectionof G, and k is an integer such that M is separated by every k elementsof G — Go, then G is upper semi-continuous and X' is locally connected.

Proof. Suppose the sequence of points pl9 p2, converges to p0,where p{ (i = 0,1, •) is a point in a continum gt of the collection G.It will be shown that g0 z> lim sup {#{}. If lim sup {&} intersects eachcontinuum of an uncountable subcollection Gλ of G, then, as in Theorem1, obtain an uncountable collection Z of cuttings of M, each being thesum of k continua of Gx — Go, and no one containing gQ. By Theorem

1148 H. C. WISER

W, there is an element Q of Z such that M — Q is the sum of twomutually separated sets Hλ and H2 containing two continua g and g\respectively, of Gλ — Go and such that Hx Z) g0. But there exists aninteger n such that H± D gt for all i > n; thus gf cannot intersect limsup {g%). This contradiction implies that g0 3 lim sup {#;} and G is uppersemi-continuous.

If X' were not locally connected there would exist a sequence ofdisjoint nondegenerate continua Xlf X2, in Xf converging to a non-degenerate continuum Xo in X'. Let F be the collection consisting ofG and the individual points of M — G*. The collection F i s upper semi-continuous, and ΛF is connected. The nondegenerate continuum f~\X0)contains an uncountable collection of mutually exclusive cuttings of M,each consisting of k elements of G — Go; thus Xo contains an uncountablecollection of mutually exclusive cuttings of M', each consisting of kpoints. This contradicts Corollary W; hence X' is locally connected.

THEOREM 3. (a) If G is a nondegenerate collection of disjointcontinua filling a continuum X on a 2-manifold, and Go is a countablesubcollection of G such that every continuum of G — Go separates- themanifold, then G is upper semi-continuous and Xf is a dendron.

(b) If each element of G separates the manifold into two com-plementary domains, then Xr is an arc.

Proof of (a). From Theorem 2, G is upper semicontinuous, and X'is locally connected. From the proof of Corollary 1.1, all but a countablenumber of elements of G separate X; thus Xf has at most countablymany nonseparating points. Every nondegenerate subcontinuum of X'then contains uncountably many separating points of Xr so that X' isa dendron [17, (1.1), p. 88].

Proof of (b). For each point x in X', let gx = f~\x). Supposethat some complementary domain D of X has a boundary which in-tersects two elements ga and gb of G. Since X' is a dendron by (a),there exists an arc [ab] in Xf. Let D1 be the complementary domainof f~\[ab\) which contains D. By Corollary 1.1, there is a point c of(ab) such that gc does not intersect the boundary of any complementarydomain of f~\[ah\). Let the two complementary domains of gc be Hand K, with A lying in H. Since D lies in H, ga and gb together withf-\[ac)) and f"\{cb\) must lie in H. All complementary domains off-\[ab\) must then lie in H, and K is empty. From this contradictionwe conclude that the boundary of any complementary domain of X mustlie in one element of G.

If g is an element of G which contains the boundary of some com-plementary domain of X, then as in a proof of Cohen's [7, Lemma 2.3],


it can be shown that g does not separate X. As in further proofs ofCohen's [7, Lemma 4.2 and Lemma 4.3], if [cd] is an arc of X' and pa point of the open arc {cd), then gv must separate gc from gd in M;thus, Xr cannot contain a simple triod. From part (a) above, Xf is adendron; therefore, Xf is an arc.

THEOREM 4. If G is a nondegenerate collection of disjoint continuafilling a plane continuum X such that each element of G separatesthe plane into two complementary domains, then there exist two ele-ments g0 and g1 of G such that X — (g1 + g0) is an open annulus.

Proof. By Theorem 3, Xr is an arc. From the proof of Theorem3, no element of G containing the boundary of a complementary domainof X can separate X. Using Theorem 1 and proceeding as in the casewhere G is a collection of simple closed curves [7, Theorem 4], we mayshow that the boundary of the unbounded complementary domain of Xmust be contained in an element gx of G corresponding to an end pointof X\ the element g0 of G corresponding to the other end point of Xr

must lie in the interior complementary domanin of gu and every pointcommon to the interior domain of g1 and the exterior domain of g0 mustbe in X.

THEOREM 5. If G is a collection of disjoint continua fillinga plane continuum such that each element of G separates the planeinto two complementary domains and is irreducible with respect toseparating the plane, then G is a continuous collection.

Proof. Let X be the plane continuum filled by G. As in Theorem 3and 4, G is upper semi-continuous, X' is an arc [ab], and the interiorof X is an open annulus. For each x in [ab] let gx be the element ofG such that f(gx) = x. It will be sufficient to show that no sequenceof elements of G converges to a proper subset of an element of G.

Suppose there is a sequence gH, gX2, of elements of G convergingto a proper subset h of an element gXQ of G. Suppose without loss thatx0 Φ a. Since gXQ is irreducible with respect to separating the plane,there exists an open disk D containing h but not containing all of gXQ

and such that D ga = φ if x0 = b and D (ga + gb) = φ if x0Φ b. Forsome x in [ab], gx lies entirely in D. Since x is neither a nor b, thereexists a sub-arc [cd] of (ab) such that f~~ι([cd]) is contained in D andx is in (cd). From the proof of Theorem 4, f~\(cd)) is an open annuluswith inner boundary contained in gc (or gd). Ύhenf'^ac]) (or f~\[db]flies in D, contradicting the choice of D.

Note, In the following theorem, we are justified in referring to

1150 H. C. WISER

X', since Theorem 2 and Theorem 3 assure us that G is upper semi-continuous.

THEOREM 6. If G is a collection of disjoint simple closed curvesfilling a continuum X on a 2-manifold M such that Xr is an arc,then X is an annulus, a Mobius strip, or a Klein bottle.

LEMMA 6.1. If G is a collection of disjoint simple closed curvesfilling a continuum X on a 2-manifold M such that X' is an arc ora simple closed curve, then G is a continuous collection.

Proof of Lemma 6.1.

Case 1. Suppose Xf is an arc. The proof proceeds in the samefashion as the proof of Theorem 5, since a proper subset h of an ele-ment of G is an arc or a point, and each open set containing h containsan open disk containing h.

Case 2. Suppose Xr is a simple closed curve, and g19 g2, is asequence of elements of G converging to a proper subset h of an elementgf of G. We may break X' into two arcs Ax and A2 with f{gr) interiorto Ax. We may then choose a subsequence of gu g2, whose elementscorrespond to points in Ax and, considering this subsequence and the arcAlf proceed as in Case 1.

LEMMA 6.2. Under the hypotheses of Theorem 6, X is a 2-manifoldwith boundary.

Proof of Lemma 6.2. Let X' be the arc [ab], and for each x in[ab] let gx be the element of G such that f(gx) = x. By covering gx

with a circular chain of open disks, an open set of M containing gx

may be obtained which is homeomorphic to an open annulus or an openMobius strip.

Case 1. Suppose gx lies in an open annulus R, and x is in (ab);then there is an arc A of Xf such that f~\A) is in R and x is an in-terior point of A. Using a theorem of Cohen's [7, Theorem 4], f~\A)is a closed annulus, and each point of gx is contained in an open diskin X.

Case 2. Suppose gx lies in an open annulus R, and x is an endpoint(say b) of Xf\ then, as in Case 1, there exists a closed annulus Rλ con-tained in R such that Rx = f~\[cb]), where [cb] is a subarc of Xf. Let


p be a point of gx, and D be a Euclidean neighborhood of p in R suchthat the diameter of D is less than the distance from gx to f~\Xf —(cb\). Then D X is contained in JBlf and considering X as space, p hasa neighborhood in D X whose closure is homeomorphic to a closeddisk.

Case 3. Suppose #x lies in an open Mobius strip R, but not in theinterior of any annulus. Suppose without loss that x Φ a. If gx sepa-rates R then R — gx = H + K where H is an open annulus and K anopen Mobius strip, or H is an open disk and K an open Mobius stripwith a closed disk removed. In either case K contains a simple closedcurve J which fails to separate K, and thus fails to separate R; thengx is contained in the open annulus R — J. Thus gx does not separateR, and R — gx is an open annulus. Let [ex] be a subarc of [ab] suchthat f~\[cx)) is in 12 — gx; then /^flcx)) is a half open annulus Rx from[7, Theorem 4]. By Lemma 6.1, G is continuous, and the boundary ofRx must be the sum of gc and gx. If x φ b there is another half openannulus R2 = /"^((OHZ]) in R — gx whose boundary is the sum of gx andgd. But then gx would lie interior to the annulus f~\[cd]). This con-tradicts the choice of gx; thus x must be equal to b. Let p be a pointof gb. Choose a disk Rp which contains p, has a simple closed curveC for a boundary, does not intersect f~\[ac\), and is such that the sumof gh cl(Rp) and C is a theta-curve. Let Rx and 2?2 be the two com-plementary domains of the theta-curve which lie in Rp. Since p is onthe boundary of f~\[cb)), Rx (say) must intersect f~\[cb)). If Rx — X isnot empty, R± must intersect the bonndary of f~\[cb)) since R± is con-nected. But neither gc nor gϋ intersect Ru and thus R± is containedin X. Similarly, either R2 does not intersect X, or R2 lies entirely inX; in either case, considering X as space, p has a neighborhood inX Rp whose closure is homeomorphic to a closed disk.

Thus, in any case, X is a 2-manifold with boundary.

Proof of Theorem 6. It follows from results of J. M. Slye's [15,Theorem 1 and Corollary 10] that if G is an upper semi-continuouscollection of simple closed curves filling a continuum X which is a 2-manifold with boundary, and Xf is an arc, then X must be an annulus,a Mobius strip or a Klein bottle. Thus, Theorem 6 follows directlyfrom Lemma 6.2 and Slye's results.

REMARK. The following is a brief outline of a direct proof ofTheorem 6, which does not use Slye's results.

In Case 3 of the proof of Lemma 6.2, cover gb with a set of opendisks Ru R2, , Rn in M — f~\[ac\) with boundaries consisting of simpleclosed curves Clf C2, , Cn such that R1 + R2 + + Rn is an open

1152 H. C. WISER

Mobius strip and such that, for i = 1, 2, , n, the sum of C< andgb cl(Ri) is a theta-curve with interior complementary domain Hi and K{.Suppose that the complementary domains have been numbered so thateach domain in the sequence Hlf H2, , Hn, Ku K2, , Kn, H± intersectsthe next domain in the sequence. As before, any domain intersectingX must lie in X; thus by an inductive process the whole Mobius stripRx + i?2 + + JR« lies in X. If c is in (αδ), it can be shown that,depending on whether ga and gb fall under Case 2 or Case 3, each off'\[ac\) and f~\[cb]) is an annulus or a Mobius strip. Thus X itselfmust be an annulus, a Mobius strip, or a Klein bottle.

COROLLARY 6.1. If G is a collection of disjoint simple closedcurves filling a continuum X on a 2-manifold M such that X' is a simpleclosed curve, then X must fill M and be a torus or a Klein bottle.

THEOREM 7. If G is a nondegenerate collection of disjoint simpleclosed curves filling a proper subcontinuum X of a 2-manifold M, thenX must be an annulus or a Mobius strip.

Proof. By Theorem 2 and Theorem 13, G must be upper semi-continuous and Xf locally connected. Thus Xf must be a dendron, forif Xf contains a simple closed curve, then, by Corollary 6.1, X wouldfill M. Since M is a 2-manifold, Theorem 6 implies that X' contains nosimple triod. Thus Xr must be an arc, and Theorem 7 follows fromTheorem 6.

REMARK. If the restriction that X be a proper subcontinuum ofM were removed in Theorem 7, X could also be a torus of a Kleinbottle.

COROLLARY 7.1. If G is a collection of disjoint simple closedcurves filling a continuum X on a 2-manifold M, then G is continuousand X' is an arc or a simple closed curve.

Proof. As in the proof of Theorem 7, either the decompositionspace X' is an arc or it contains a simple closed curve. If X' containsa simple closed curve, it must be one by Corollary 6.1. The continuityof G follows from Lemma 6.1.

4. Conditions under which a homogeneous continuum on a 2*manifold is a simple closed curve*

THEOREM 8. If X is a homogeneous proper subcontinuum of a 2


manifold M, and X contains a simple closed curve, then X must be asimple closed curve.

Proof. Suppose X is not a simple closed curve.(1) X is one-dimensional, for otherwise X would contain an open

set in M and thus would contain M. This contradicts the hypothesisof Theorem 8.

(2) X is not locally connected, for using (1) and a result of An-derson's [1, Theorem 13], X must be either a simple closed curve orthe universal one-dimensional curve. This is a contradiction in eithercase, since the universal curve contains no open set which can beembedded in the plane.

(3) Because X is not locally connected, there is a disk on M con-taining an open set of X which has uncountably many components [7,Lemma 2.1 and Corollary 2.11].

(4) Suppose X contains a simple triod. By (3), some open set Dof X is contained in a disk of M and has uncountably many components;thus the homogeneity of X implies that D contains uncountably manydisjoint simple triods. This contradicts a theorem of R. L. Moore's [13,Theorem 75, p. 254]; thus X contains no simple triod.

(5) No two simple closed curves in X intersect, for if some twodid intersect then X would contain a simple triod, contrary to (4).

(6) Let G be the collection of all simple closed curves in X; thenG fills X and the elements of G are disjoint, since by homogeneity eachpoint of X lies on a simple closed curve and by (5) no two simple closedcurves intersect.

By Theorem 7, X must be an annulus or a Mobius strip; this con-tradicts (1), and Theorem 8 follows.

COROLLARY 8.1. If a nondegenerate proper subcontinuum of a 2-manifold is locally connected and homogeneous, then it must be asimple closed curve.

Proof. This corollary follows from (1) and (2) in the proof ofTheorem 8.

COROLLARY 8.2. No locally homogeneous proper subcontinuum ofa 2-manifold contains a simple triod.

Proof. This corollary is obtained as in (3) and (4) of the proof ofTheorem 8, where the homogeneity condition may be replaced by localhomogeneity.

REMARK. Cohen [7, Theorem 3] has shown that a homogeneous,

1154 H. C. WISER

arcwise connected, plane continuum must be a simple closed curve. Ina similar fashion, we have the following result.

THEOREM 9. // the nondegenerate continuum X is arcwise con-nected, contains no simple triod, and is either nearly homogeneous orlocally homogeneous; then X is a simple closed curve.

COROLLARY 9.1. If a nondegenerate proper subcontinuum of a 2-manifold is locally homogeneous and arcwise connected then it must bea simple closed curve.

This corollary follows from Corollary 8.2 and Theorem 9.

THEOREM 10. If a nondegenerate proper subcontinuum of a 2-manifold is aposyndetic and homogeneous, then it must be a simpleclosed curve.

LEMMA 10.1. Suppose A is an arc, and G is a countably infinitecollection of disjoint arcs such that if A{ = \x{yλ is an arc of G fori — 1, 2, , then A{ A — x{ + y%\ then for any positive integer k,there exist k disjoint simple closed curves contained in A + G*.

Proof of Lemma 10.1. For convenience, let A be the unit interval[01]. Without loss of generality, suppose that xl9 x29 is a monotonesequence converging to a point x of A, and y19 y2, is a monotonesequence converging to a point y of A.

Case 1. If x Φ y, let Ix and Iy be disjoint open intervals (or halfopen intervals if x or y is an endpoint of A) of A, containing x and yrespectively. Suppose, without loss, that each point of the sequencexl9 x2, lies in Ix and each point of the sequence yl9 y2, lies in Iy.Let [pq]A denote a subinterval of A with the points p and q as end-points. Then J l i 2 = [XχX^A + [y{y^A + Λ + A2 is a simple closed curvein A + G*. Indeed, {J{2n-i),2n} for n = 1, 2, , fc is a set of k disjointsimple closed curves in A + G*, where J2n-1,2n — [^Π-I^ΛA + [v^n-iViΛA+ A2n~ι + A2n.

Case 2. Suppose x — y. If the two sequences xu x2, and yl9

y2, converge to x from opposite sides, then the construction in Case1 will give the desired set of simple closed curves. Suppose for con-venience, that both sequences converge to x from the left. Thereexists an increasing sequence of positive numbers r19 r2, such thatfor each i, both xri+i and yu+1 lie to the right of both xH and yH onA. Then {Jn}9 for n = 1, 2, •••, k, is a set of k disjoint simple closed

curves in A + G*, where Jn = [xrnVrn]A + Arn.

LEMMA 10.2. If X is a continuum satisfying the hypothesis ofTheorem 10, then the boundary of each complementary domain of Xis locally connected.2

Proof of Lemma 10.2. Let D be a complementary domain of X.The boundary of D must consist of a finite number of continua B19 B2,• , Bm by a lemma of Roberts' and Steenrod's [14, Lemma 1]. SupposeBx fails to be locally connected at a point q. Let R be a disk contain-ing q such that cZ(i2) intersects no B{ for i = 2,3, •••, m. By a stand-ard construction, there are two open sets Rx and R2 with closures inR, a continuum Xo in i2, and a sequence of disjoint continua Xlf X2f

in i? with the following properties:(1) CZ(JRI) does not intersect cl{R2),(2) i?i and iu2 have simple closed curves d and C2, respectively,

for boundaries,(3) for each i, JSΓ contains both a point of CΊ and a point of C2

and is a component of the common part of Bx and R — (Rx + R2), and(4) the sequence Xu X2, converges to Xo.Let p be a point of Xo — {cl(R^) + cl(R2))f ε be a positive number

less than the distance from p to cl(R^ + cl(R2), and Vs(p) be a circularneighborhood of p with a circle Cs as boundary. Then there exists acircular neighborhood V8(p) with a circle Cδ as boundary such thatδ < ε and all of X— V2(p) lies in one component N of X— V5(p). Thatsuch a Fδ(p) exists follows as in a theorem of Whyburn's [16, (6.22)],since X, being compact and aposyndetic, must be semi-locally connected,and p must not be a cut point of X because X is homogeneous. With-out loss of generality, suppose that the X{ (i = 1, 2, •••) have beenchosen so that each intersects V&(p) and such that (Xx Cj), (X2 Cj),

• are ordered, as named, along the simple closed curve Cj (j = 1, 2).Without change of notation, consider Xt (i = 1, 2, •) to be irreduciblefrom d to C2.

An open set Oλ in JB is bounded by Xx + X2 + An + Au, where An

is an arc in Ct irreducible from Xx to X2 and intersecting no Xs withj > 2, and A12 is a similar arc in C2. In the same way, for i = 1, 2,obtain a ' 'corridor'' 0^ between the continua Xi and Xi+1 bounded byXi9 Xi+1 and arcs Aa and Ai2 in Cx and C2 respectively. Now, sinceeach Xi is on the boundary of D, for i = 2, 3, 4, we may choose apoint Zi in X; Vβ(p) and a neighborhood Ϊ7< of z{ in y"δ(ί>) such thatUi contains a point ^ of D and intersects no Xά for j Φ i. The point

2 As noted by the referee for this paper, Lemma 10.2 is closely related to results(mainly for the plane or ^-sphere) of Jones' [9], Whyburn's [16], and Wilder's [18 and 19].Indeed, the proof given here was motivated by the proof of Theorem 14 in [16].

1156 H. C. WISER

Pi must lie in 0^ or O ^ . Possibly discarding some of the Pi (ί = 1,2, •••) and X{ (i = 2, 3, •••), and re-numbering the remaining points,continua, and corresponding corridors (retaining the same order as before);we arrive at a set of points {pj of D with fpi in Oi Vδ(p) (ί — 1, 2, •)•

Run an arc [PιP2] in Z) from ^ to p2. Let ^x be the last point ofCδ Oi on [PiPa] in the order px to p2. Let yλ be the first point of Cδ

in the order xλ to p2 along (ccxpj, a subarc of [PiP2]. Then ^ is in someOjl with yx =£ 1, and (x^) lies in J9 — cί(Fδ(p)), where (x^) is an opensubarc of [pλp2]. Choose n2 such that n2 > 1 and w2 > j l β Now run anarc [pn2pn2+1] in D — [xλx2\ from p%2 to pΛ a + 1. As before let x2 be thelast point of Cδ 0%2 on [p%2p%2+i] in the order p%2 to pn a + 1, and y2 thefirst point of Cδ in the order #2 to p%2+1 along (α?2pW2+J; then y2 is insome O ia with j2φn2, and (£c23/2) lies in D-cl(V5(p)). Continue con-structing disjoint arcs [^^] in this manner such that for i — 1, 2, :

(1) Xi is in Cδ 0% i,(2) y{ is in Cδ 03 t with ^ ^ n<,(3) nk > Ui ίor k > i and w > jt for h > i,

(4) [ίc^] lies in Z) - Σ^lt^^/.], and(5) (xiVi) lies in 2?-rf(Fδ(p)).

The set ^7=i(xi + 2/ί) is a subset of an arc A in Cδ. Lemma 10.1may now be applied to the arc A and the set of arcs {Aτ) where, fori = 1, 2, , Ai = [XiVi], Using Theorem 13 and the construction inLemma 10.1, obtain m disjoint simple closed curves J[, J2, , J'm whosesum separates M and such that each one is the sum of one or twoarcs from the set {AJ and one or two arcs of Cδ.

Case 1. Suppose the simple closed curves are of the form givenin Case 1 and the first part of Case 2 of Lemma 10.1. We can thenre-number the arcs and corridors so that J'n = [a^-Ajδ + (j^-i^Js +A2n-λ + A2n (n = 1, 2, , m), f and xlf x2, , x2m forms an ordered setalong Cβ, where [pg]δ denotes a subarc of A with endpoints p and g.For each ^ (^ = 1, 2, , m), replace the arcs [x^^x^s and [y^-^nhwith arcs [sca»-i#2*]r and [y2n-iy2n]v> respectively, such that the 2m arcsof ([Xm-iXinlrf [V2n-iy2n]v} are disjoint, and (x2n-1x2n)r and (y2n-xy27)v are

open arcs lying in Fδ(p). Let Jlf ,Jn be the new simple closedcurves obtained from J[, , J'm by the replacement of arcs as describedabove.

We will now show that if zx and z2 are points of Cδ which lie indifferent corridors Ox and O2, respectively, and each of zx and z2 is anend point of some arc in D like the [x^i] described above; then thearc [^^2]δ of Cδ must intersect N. Let [zλz3] be an arc in D such that23 is in Cδ Oi where iψl and (zλz^) lies in Z) — cl(V8(p)). Let [^4]be an arc in D such that z± is in Cδ Ojf where j Φ 2 and (z2z4) lies in


D — cl(Vδ(p)). Let B be the subarc [zλz[] of [zfo], such that z[ is thefirst point of Cε on [z^] in the order zλ to 23. Let E be the subarc[z2z'2] of [£224L such that z[ is the first point of Cε on [2224] in the orderz2 to 24. Let C = [2lz2]ε be an arc of Cε in the same direction as thearc F = [zλz2]δ on Cδ. Let J be the simple closed curve B + C + E +i*7 whose interior lies between Cδ and Cε. Then i*1 must intersect Xsince ^ and z2 lie in different corridors, and B and i? are in D. SupposeN does not intersect F, then no component of X — Vδ(p) can intersectboth F and C, for such a component would be contained in N. How-ever, every component of X— Vδ(p) must intersect Cδ, and thus eachcomponent of X — Vδ(p) which intersects the interior of J must in-tersect either F or C. Let H be the set of components of X — Vδ(p)intersecting F, and let K be the set of components intersecting CThen H* and K* are disjoint closed sets. Then by a theorem provedby Moore [13, Theorem 12, p. 189], there exists an arc from B to E,lying interior to J except for end points, which does not intersect Xand thus must lie in D. But then zx can be connected to z2 by an arcin D cl(Vζ(p)); this contradicts the choice of ε and the fact that zx

and z2 lie in different corridors Oλ and O2. Therefore N must intersectthe interior of F = [£i22]δ.

By the construction of Lemma 10.1, all of the [^2w-i^2n]θ, for n =1, 2, •••, m, must lie in an arc of Cδ containing none of the Vι (i = 1,2, « ,2m). Then from the discussion above, since xlf x2, and xs eachlie in different corridors, the interior of each of the arcs [xxx^δ and[cc2x3]δ must contain a point of N. We can then choose an arc {nxx2u2\h

of Cδ such that

(1) ux + u2 c iV,

(2) (%i£awa)8 does not intersect N,(3) %! is a point of (α?1α?a)δ,(4) u2 is a point of (#2#3)δ, and(5) [ttx&a la — x2 intersects no Jn (n = 1, 2, , m).

Case 2. Suppose the simple closed curves J[, , J'm are of theform given in the second part of Case 2 in the proof of Lemma 10.1.We can then re-number the arcs and corridors so that Jf

n — [xnVnh +An (n = 1, 2, , m), and xl9 y19 x2, y2, , xm, ym forms an ordered seton A. As before, for n — 1, 2, •••, m replace [&%]/w]δ by the arc [XnVn]vsuch that (£C«2/Λ)Γ lies in Vδ(p); thus obtain the set of disjoint simpleclosed curves {Jn}, where Jn = [xnyn]v + An.

Since xλ and y± lie in different corridors as do yλ and 05a, we canobtain an arc [u^u^ of Cδ such that

(1) fa + ujczN,(2) (^i^/i^2)δ does not intersect JV,

1158 H. C. WISER

(3) ux is a point of(4) u2 is a point of {yλx2)^ and(5) [uû^ — yλ intersects no Jn (n = 1, 2, , m).Notice that, in each of the above cases, N does not intersect

Jx + J2 + + Jm.For purposes of the remainder of the proof, Case 1 and Case 2 are

identical, and we will use the notation of Case 2. Assume, withoutloss, that Jx + J 2 + + Jm separates M, but J 2 + + Jm does not.There exist points c and d separated from each other on M by Jx + J2

+ * * + Jm and an arc [cd] of M which does not intersect J2 + +Jm and thus must intersect Jx. Choose εx > 0 such that εx < min [p(u19

Jι), p{u2, JO, ρ(c, Jλ), ρ(d, Jx), ρ(Jlf Jk), k = 2, 3, , m]. Let ?7 be anannulus or Mδbius strip contained in an ex cover of Jx such that Jx isinterior to U. Let Ci be the first point of Jx on [cd] in the order c tod and c0 a point of U [ccί] preceding cx on [cd] in the order c to d.Then in U — Jλ there is an arc Bλ from c0 to a point αx of {uxy^)5 (con-sider %! and u2 re-numbered if necessary). Similarly construct an arcB2 in U — Jλ from d0 to a point ί of (yxu^ or (u^)^ where d0 is apoint of 17 [cd] preceding the first point d4 of J x on [cd] in the orderd to c. If 6X is in {yxu^ then [cc0] + Bx + [^^^5 + iV + [δi^2]δ + B2 +[ώoώ] is a continuum in M - ΣΓ=]/; containing both c and d, and if bλ

is in (wyî then [cc0] + JBX + [^αjβ + N + [îδjβ + # 2 + [dod] is a con-tinuum in I - ΣaT=iJi containing both c and c£. This contradictionestablishes Lemma 10.2.

Proof of Theorem 10. Suppose X is an aposyndetic homogeneousnondegenerate subcontinuum of a 2-manifold, and X is not a simple closedcurve. By Theorem 8, X contains no simple closed curve. It followsfrom Corollary 8.2 and Lemma 10.2 that a component Bx of the bounda-ry of a complementary domain D of X must be an arc. Cover Bx withan open disk R that does not intersect any other component of theboundary of D and does not contain X. Since X is connected, B mustintersect X — Bx. Since Bx is part of the boundary of D, R — Bx is aconnected set intersecting both X and D, and thus intersecting thebounday of D\ this contradicts our choice of R. Therefore X is a simpleclosed curve.

COROLLARY 10.1. Every 2-homogeneous nondegenerate proper sub-continuum of a 2-manifold is a simple closed curve.

Proof. C. E. Burgess has shown that any 2-homogeneous continuumis aposyndetic [6, Theorem 7]. Corollary 10.1 then follows directly fromTheorem 10 and the fact that each 2-homogeneous continuum is homo-geneous [5, Theorem 1].


COROLLARY 10.2. Every homogeneous nondegenerate continuum,which contains a non-cutpoint and lies on a 2-manifold, is a simpleclosed curve.

Proof. This follows directly from Theorem 10 and the proof of atheorem of Jones' [10, Theorem 2].

5* Decomposition of decomposable homogeneous continua*

THEOREM 11. If a proper subcontinuum X of a 2-manifold M isdecomposable and homogeneous, then there exists a continuous collectionG of disjoint continua filling X such that Xr is a simple closed curve,and the elements of G are mutually homeomorphic, homogeneous tree-like continua.

Proof. A theorem of Jones' [12, Theorem 1] gives a nondegeneratecontinuous collection G of mutually exclusive continua filling X suchthat

(1) X' is a homogeneous aposyndetic continuum,(2) the elements of G are mutually homeomorphic, homogeneous

continua, and(3) if g is a continuum of the collection G and K a subcontinuum

of X containing both a point of g and a point of X — g, then g is asubset of K.

Case 1. Suppose that each element g of G is treelike. A theoremof Roberts' and Steenrod's [14, Theorem 1] implies that the collectionconsisting of the elements of G together with the individual points ofM — X forms an upper semi-continuous decomposition of M such thatM' is homeomorphic to M. Then since X' is an aposyndetic homogene-ous continuum on a 2-manifold, it must be a simple closed curve byTheorem 10.

Case 2. Suppose that each element of G is nontreelike. Since X'is homogeneous, it can have no separating point; thus, for any elementg of G, X — g is connected and lies in a complementary domain D ofM—g. By a result due to Roberts and Steenrod [14, Lemma 1], Dmust contain a continuum K such that D — K = Hx+ + H8 wherethe Hi are disjoint open cylinders. By the continuity of G, there issome subcollection Gx of G filling a continuum A in one of these opencylinders. Think of this cylinder as embedded in the plane. Eachelement of Gλ must separate the plane [2, Theorem 6]; indeed, eachelement of Gx must have two complementary domains, since the elementsof G are homeomorphic and the plane does not contain uncountably

1160 H. C. WISER

many disjoint continua each having three or more complementary domains.By Theorem 4, A is two-dimensional; this is a contradiction, since X isone-dimensional. Case 2 is thus vacuous, and Theorem 11 is established.

REMARK. In the proof of Theorem 11, each element of G± fails toseparate the plane, and thus each element of G is indecomposable bya theorem of Jones' [11, Theorem 2]. The indecomposability of theelements of G also follows, as in the proof of Theorem 12, from atheorem proved by E. Dyer [8, p. 591],

THEOREM 12. If X is a decomposable continuum which is homo-geneous and hereditarily finitely multicoherent3, then there is a non-degenerate continuous collection G of disjoint continua filling X suchthat X1 is a simple closed curve and the elements of G are mutuallyhomeomorphic, homogeneous, indecomposable continua.

LEMMA 12.1. An aposyndetic hereditarily finitely multicoherentcontinuum must be locally connected.

Proof of Lemma 12.1. Let Xbe an aposyndetic hereditarily finitelymulticoherent continuum. As in the proof Burgess [6, Theorem 8] hasgiven for the case where X is hereditarily unicoherent, for any subcon-tinuum K oΐ X and point p of X -— K, there exists a positive integerm and continua X19 X2, , Xm, Ylf Y2, , Ym such that K c [(X — X±)+ (X - X2) + + (X - Xm)] and, for each i (i ^ m), X, + Y, = Xand p c X — Yit Since X is hereditarily finitely multicoherent, thecommon part of the continua Xlf X2y , Xm is the sum of a finitenumber of continua Zu Z2, Zn not intersecting K. Then X — Zx +Z2 + ••• + Zn + Yχ+ Y2 + + Ym, and X is locally connected by atheorem of Moore's [13, Theorem 51, p. 134].

LEMMA 12.2 A locally connected, hereditarily finitely multicoherentcontinuum X must be hereditarily locally connected.

Proof of Lemma 12.2. Suppose 7 is a subcontinuum of X whichfails to be locally connected. Then there exists a sequence of disjointnondegenerate continua Nu N2, in Y converging to a nondegeneratecontinuum N. Let px and p2 be points of N and Rλ and R2 be opensets containing, respectively, pι and p2 such that cl(R^ cl{R2) — 0.Since X is locally connected, there exist connected open sets Ui and

3 A continuum X is said to be finitely multicoherent if for any two subcontinua Xι andXi of X such that X = Xi + X2, the common part of Xi and X2 is the sum of a finitenumber of continua. If every subcontinuum of X is finitely multicoherent, then X is saidto be hereditarily finitely multicoherent.


U2 containing px and p29 respectively, such that Ux and U2 are con-tained in i?i and R2, respectively. We may choose a sequence of disjointcontinua M19 M2, such that each Mt is a subcontinuum of some Nj9

each Mi is irreducible from cZ(ίTi) to cl(U2), and the sequence M19 M2,converges to a subcontinuum M of N. Then cl( £7ί) + ΣΓ=i^ίί + M andcK U2) + ΣΓ=i^; + M are two subcontinua of X whose intersection is thesum of an infinite number of disjoint continua. This is a contradiction;hence, X must be hereditarily locally connected

Proof of Theorem 12. There exists a nondegenerate continuouscollection G of disjoint continua filling X having the properties givenat the beginning of the proof of Theorem 11. If A is a subcontinuumof X', a result of Whyburn's [17, p. 154] for monotone decompositionsimplies that A is finitely multicoherent if f~\A) is finitely multicoherent;thus Xf must be hereditarily finitely multicoherent. But X' is apo-syndetic and thus, by Lemma 12.1, must be locally connected. ByLemma 12.2, X' is hereditarily locally connected, and Burgess [6,Theorem 14] has shown that a nondegenerate continuum which is homo-geneous and hereditarily locally connected must be a simple closedcurve.

To show the indecomposability of the elements of G, let us choosea continuous subcollection Gx of G which is an arc with respect to itselements. Let g0 and g1 be the end elements of this arc, and x0 andxx be points in g0 and gx respectively. A continuum K in G* whichcontains xQ and x± must also contain each element of Gλ which it in-tersects. But K must then contain both g0 and gl9 and, since K isconnected, K must fill Gf; i.e. G* is irreducible from x0 to x±. Dyer[8, p. 591] has shown that such a collection must contain an inde-composable continuum; thus, each continuum of G is indecomposable.

REFERENCES

1. R. D. Anderson, One dimensional continuous curves and a homogeneity theorem, Ann.of Math., 68 (1958), 1-16.2. R. H. Bing, Snake-like continua, Duke Math. J., 18 (1951), 653-663.3. , A simple closed curve is the only homogeneous bounded plane continuum thatcontains an arc, Canadian J. Math., 12 (1960), 209-230.4. R. H. Bing and F. B. Jones, Another homogeneous plane continuum, Trans. Amer. Math.Soc, 9O (1959), 171-192.5. C. E. Burgess, Some theorems on n-homogeneous continua, Proc. Amer. Math. Soc, 5(1954), 136-143.6. , Continua and various types of homogeneity, Trans. Amer, Math. Soc, 88(1958), 366-374.

7. H. J. Cohen, Some results concerning homogeneous plane continua, Duke Math. J., 18(1951), 467-474.8. E. Dyer, Irreducibility of the sum of the elements of a continuous collection of continua,Duke Math J., 20 (1953), 589-592.

1162 H. C. WISER

9. F. B. Jones, Aposyndetic continua and certain boundary problems, Amer. J. Math., 63(1941), 545-553.10. . A note on homogeneous plane continua, Bull. Amer. Math. Soc, 55 (1949),113-114.11. , Certain homogeneous unicoherent indecomposable continua, Proc. Amer. Math.Soc, 2 (1951), 855-859.12. , On a certain type of homogeneous plane continuum, Proc. Amer. Math Soc,6 (1955), 735-740.13. R. L. Moore, Foundations of Point Set Theory, Amer. Math. Soc Colloquium Public-ations, vol. 13, 1932.14. J. H. Roberts and N. E. Steenrod, Monotone transformations of two-dimensional mani-folds, Ann. of Math., 39 (1938), 851-863.15. J. M. Slye, Collections that are 2-manifolds, Duke Math. J., 24 (1957), 275-298.16. G. T. Whyburn, Semi-locally connected sets, Amer. J. Math., 61 (1939), 733-749.17. , Analytic Topology, Amer. Math. Soc. Colloquium Publications, vol. 28, 1942.18. R. L. Wilder, Sets which satisfy certain avoidability conditions, Casopis pro PestovaniMatematiky a Fysiky, 67 (1938), 185-198.19. , Property Sn, Amer. J. Math., 61 (1939), 823-832.

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Pacific Journal of MathematicsVol. 12, No. 3 March, 1962

Alfred Aeppli, Some exact sequences in cohomology theory for Kählermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791

Paul Richard Beesack, On the Green’s function of an N-point boundary valueproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

James Robert Boen, On p-automorphic p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813James Robert Boen, Oscar S. Rothaus and John Griggs Thompson, Further results

on p-automorphic p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817James Henry Bramble and Lawrence Edward Payne, Bounds in the Neumann

problem for second order uniformly elliptic operators . . . . . . . . . . . . . . . . . . . . . . . 823Chen Chung Chang and H. Jerome (Howard) Keisler, Applications of ultraproducts

of pairs of cardinals to the theory of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835Stephen Urban Chase, On direct sums and products of modules . . . . . . . . . . . . . . . . . . . 847Paul Civin, Annihilators in the second conjugate algebra of a group algebra . . . . . . . 855J. H. Curtiss, Polynomial interpolation in points equidistributed on the unit

circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863Marion K. Fort, Jr., Homogeneity of infinite products of manifolds with

boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879James G. Glimm, Families of induced representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885Daniel E. Gorenstein, Reuben Sandler and William H. Mills, On almost-commuting

permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913Vincent C. Harris and M. V. Subba Rao, Congruence properties of σr (N ) . . . . . . . . . . 925Harry Hochstadt, Fourier series with linearly dependent coefficients . . . . . . . . . . . . . . . 929Kenneth Myron Hoffman and John Wermer, A characterization of C(X) . . . . . . . . . . . 941Robert Weldon Hunt, The behavior of solutions of ordinary, self-adjoint differential

equations of arbitrary even order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945Edward Takashi Kobayashi, A remark on the Nijenhuis tensor . . . . . . . . . . . . . . . . . . . . 963David London, On the zeros of the solutions of w′′(z)+ p(z)w(z)= 0 . . . . . . . . . . . . . 979Gerald R. Mac Lane and Frank Beall Ryan, On the radial limits of Blaschke

products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993T. M. MacRobert, Evaluation of an E-function when three of its upper parameters

differ by integral values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999Robert W. McKelvey, The spectra of minimal self-adjoint extensions of a symmetric

operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003Adegoke Olubummo, Operators of finite rank in a reflexive Banach space . . . . . . . . . . 1023David Alexander Pope, On the approximation of function spaces in the calculus of

variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029Bernard W. Roos and Ward C. Sangren, Three spectral theorems for a pair of

singular first-order differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047Arthur Argyle Sagle, Simple Malcev algebras over fields of characteristic zero . . . . . 1057Leo Sario, Meromorphic functions and conformal metrics on Riemann surfaces . . . . 1079Richard Gordon Swan, Factorization of polynomials over finite fields . . . . . . . . . . . . . . 1099S. C. Tang, Some theorems on the ratio of empirical distribution to the theoretical

distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107Robert Charles Thompson, Normal matrices and the normal basis in abelian

number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115Howard Gregory Tucker, Absolute continuity of infinitely divisible distributions . . . . 1125Elliot Carl Weinberg, Completely distributed lattice-ordered groups . . . . . . . . . . . . . . . 1131James Howard Wells, A note on the primes in a Banach algebra of measures . . . . . . . 1139Horace C. Wiser, Decomposition and homogeneity of continua on a 2-manifold . . . . 1145

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