Spectral estimates of Cauchy's transform inL 2(?)

Integr Equat Oper Th 0378-620X/92[060901-1951.50+0.20/0 Vol. 15 (1992) (c) 1992 Birkh~user Verlag, Basel

SPECTRAL ESTIMATES OF CAUCHY'S TRANSFORM IN L2(~)

J. Arazy and D. Khavinson 1

We prove that the singular numbers of the Cauchy transform ( C f ) ( z ) = fD l~-~dA(~) on L2(D) are asymptotically s , (C) ~ --~, while s,(CIL~(D) ) ..~ ~ (where L](D) i-s the subspace of analytic functions in L~(D)). Also, the singular numbers of the logarithmic potential ( L f ) ( z ) = fD f(~)log (t-~l~l] dA(~) on L~(D) are asympotically s , (L ) ~ ~, while sn(LJL~(D)) ..~ -~. Our methods yield the asymptotic behavior of the singular numbers

1 [ - ~ t

of the Cauchy Transform from L~(#) into L2(u) where # and u are rotation-invariant measures on I).

w I n t r o d u c t i o n .

Let ~2 C C be a bounded domain with smooth boundary. Denote by L2(~) the space of

complex-valued functions f in ~ for which the norm

[[f][2 = {f~ If(z)[~dA(z)} a/z

is finite. Here, dA(z) = d~d~, z = ( + iy, is the normal ized area measure in ~.

D E F I N I T I O N : The Cauchy Transform C = C~ is the integral operator in L2(f~) de-

fined by

(Cf)(z)= f fz(~O(dA(r As a convolution of L2-functions with the compactly supported measure ~x~dA(~), C :

L2(~) ~ L2(~t) is obviously bounded. Moreover, it is not hard to show([AKL, Thin. 2.1])

that C is in fact compact�9 This raises a question concerning the spectral picture of C or,

rather C*C (or CC*), where C* is a formal adjoint of C. In case fl = D, the unit disk, a

complete answer is given by the following theorem proved in [AKL, Thin. 2.2].

For m = 0, 1,2, . . - let

k=o k!(k + m)! �9 o o denote the m-th Bessel function of the first kind and let (3m,n)~=l be the (positive) zeros of

J,~, jm,1 < jm,2 < ' ' ' . Set

pm,n : "-2 j . . . . r e=O, 1 , . . . , n = 1 ,2 , . . . .

1The author was partly supported by a grant from the national Science Foundation.

902 Arazy and Khavinson

T H E O R E M A. [AKL, Theorem 2.2] Let f~ = D. The eigenvalues of C*C are precisely

the numbers Am,~ = Pm,~, m = 0 , 1 , . . . , n = 1 ,2 , - . . . Each eigenvaIue has multiplicity

two. An orthogonal basis for the eigenspace corresponding to ~m,~ consists of the functions

I f ,o(re,O I = } [ gin,n(reiO) = Jm+l(jm,~zr)ei(m+l) 0

This paper is an addendum to [AKL]. We apply Theorem A in order to derive a num

ber of results concerning more subtle operator-theoretic properties of operator C. In partic

ular, in w we show that C belongs to he "weak Schatten ideal" $2,~ and this is the smallest

unitary ideal containing C.

In w we study the spectral behavior of CIL~(D), where as u s u a l L~(D) = closure of

polynomials in L2(D). Remarkably, it turns out that singula r numbers {s,~}~ of PCP

and CP are asymptotically s• and hence CP, PCP 6 $1,~. (Here, P : L 2 ~ L~ is l m J m = l ,

the orthogonal projection .) In other words, C operates on analytic (or, conjugate-anMytic)

functions two times "smoother" than on the whole L 2.

In w we show how some of the previous estimates can be carried over to L2-s paces with

respect to radially symmetric measures in D. This allows one to use the est imate o brained

in w in a certain continuous scale of spaces�9

Finally, in w we treat similar questions for the logarithmic potential ope ra to r .

A c k n o w l e d g e m e n t � 9 Part of this work has been done during the first author 's visit to

the University of Arkansas at Fayetteville in June of 1990, which was part ly supported by

the Depar tment of Mathematics at the University of Arkansas.

w Ca on t h e S c h a t t e n Scale

Recall that an operator T acting from a Hilbert space HI into a Hilbert spa ce H~ is

said to belong to the Schatten ideal Sp if IITII~: = trace (T'T) p/2 < oo. Equivalently, T E Sp

provided that T is compact and the sequence of singular numbers s~(T) := An((T*T)I/~), n =

1 ,2 , . . . (counting multiplicity) belongs to gp. Then, I[Tllv = (~n s~(T)) 1/~. T is said to

belong to the "weak Sp ideal", Sp,~o, if

[ ]T l tp ,oo : = sup[sn(T)r~ 1/p] < cx3. n

[l' Ilv,~o defines a complete metric, and for 1 < p < ec, I[' II;,~ is a norm. Notice that T e S, ,~

if and only if (sn(T)) e gv,~o with I[TII~,~ = [l(sm(T))%,~. Also S~ c# Sp,~ c# n~<rSr (for details, se e [OK], IS]). The following useful result is due to B. Russo [R].

Arazy and Khavinson 903

L E M M A 2.1. Let (X, #) be a finite measure space and let

( T f ) ( x ) = i x K(x , y ) f ( y )d# (y )

be an integral operator. Let 1 < q ~ 2 and let p be the conjugate exponent : l iP + 1/q = 1.

Suppose that

J. (J. IK(x,y)lqdv(x))plqdv(y) = ~ < oz

and

J. (j. iU(x,y) l ,dv(y)) . lqdv(x) = n" < oo

Then, as an operator on L2(#), T E Sp and IITllp _< c ~ ( ~ ) 11~, (Cp is a constant).

The following lemma is elementary (cf. [AKL, proof of Theorem 2.1]).

L E M M A 2.2. Let ~ C C be a bounded set. Fix z 6 ~ and denote by Dz the disk

centered at z and of the same area as ~. Then for all q : O < q < 2, we have

dm(w) dA(w) 1

f~ r z _ wl------~ < f~= - - q/2)[A(gt)]l-q/2" - Iz -~ ~llq (1

(Here, A (~ ) = fa dA(w) = A(Dz)).

C O R O L L A R Y 2.3. C 6 Sp for all p > 2, but not to S2.

P r o o f : The fact that C is not Hilbert-Schmidt is obvious. Fix 1 < q < 2. From Lemma 2.2

we have (p is the conjugate exponent):

<_ Cq [A(rt)P/q-'/Z+l] ~/" = CqA(IT) 1/2.

By Lemma 2.1, C E Sp. []

Appealing to Theorem A, one can obtain a more precise information. For that we need

the following fact concerning the distribution of zeros of Bessel functions. It is easily duduced

from the resu l t s in [W, w

L E M M A 2.4. There exist positive constants (~,~ such that

. ( m + n) < jm.n < Z(m + n) (2.1)


for allm = 0 , 1 , . . . , n = 1 ,2 , . - . .

C O R O L L A R Y 2.5. Let f~ = D. Then C E $2,or and $2,~ is the smallest unitary ideal

containing C.

P r o o f : Let sk = sk(C). So s~ = Ak(C*C). U sing Theorem A and (2.1) we obtain 2[{(rn, n) : 2

. . . . . m > o ,n > l , m + n < ~ } l < l{k : sk > t}l < 2 l { ( m , n ) : m > O,n > l , m + n <

where I{ }1 is the cardinality of the set { }. Since ]{(re,n) : m + n <_ N}I ~ N 2, the

above inequality yields existence of cons tants a, b such that

a b - < I{,~ : s,~ > ~ } 1 < i~ t 2 - - - - _

for all t > 0 sufficiently small. Choosing t = sk we obtain

or, equivalently

a/s~ <_ ~ <_ b/s~

a 1/2 <_ kl/2sk <_ b 1/2,k = 1 , 2 , . . . . (2.2)

Hence, C E $2,~.

Since ~2,~ is the smallest symmetric sequence space containing sequence {1 /v~ ) , $2,~

is the smallest unitary ideal containing any operator T whose singular numbers s a t i s fy

sk(T) ~ 1 / v ~ . So, (2.2) implies that $2,~ is the smallest unitary ideal containing C. 1 []

In order to extend Corollary 2.5 to arbitrary f~, observe that if fh C f/2 are bounded

domains, then

Ca~ = Mnl Ca2 Mal,

where Mal = Mxa * is simply multiplication by the characteristic function of fh . As

IIMa, llmL~(a2)) = 1 , s~(Ca l ) < sn (Ca2) ,n = 1,2, . . . . Given a bounded domain f~, let D1, D2 be respectively the largest and smallest disks

contained in and containing ft. Then,

~,(cB,) _< ~ ( c a ) _< s,(CD~),n = 1 ,2 , . . . .

As s , (CD,) = Ris,(CD), where R 5 is the radius of f l fDi, j = 1,2, it follows from Corollary

2.5 that there exist positive constants a(ft) , ~(fl) such that

f l (a)/x/~ <_ s~(Ca) < a(fl)/v~. (2.3)

Hence, we obtain the following.


C O R O L L A R Y 2.6. Ca 6 $2,~ for all bounded domains ~; $2,~ is the smallest

unitary ideal containing Ca.

R E M A R K S : The above argument gives a poor insight concerning the size of the best

a(~), ~(~) in (2.3). It seems to be of interest to pursue more accurate estimates of those co

nstants in terms of geometric properties of ~.

It is not hard to extend Corollary 2.3 to unbounded domains of finite area. Wh at about

Corollary 2.6?

Finally, let us discuss the invariant subspaces of C~)CD. For A 6 T := {z : Izl = 1}

consider the rotation operator

(RAf)(z) = f(Az), f 6 L2(D).

L E M M A 2.7. For A 6 T, CRA = AR~C and C*CR~ = R~C*C.

Proof:

(CR~f)(z) -= /D f(s w

This proves the first statement. As R*~ = RX, C*R:~ = ~R~C*. Hence C*CR~ = C*AR~C =

A~R~C*C = R~C*C. []

For m 6 Z, define H,~ = { f 6 L2(D) : R~f = Amf f or all A 6 T}. Clearly, L~(D) =

Emez ~Hm.

R E M A R K : Lemma 2.7 says that C maps Hm into Hr~-I for all m 6 Z. So, C is a

"backward weighted shift". Similarly, C* maps Hm into Hm+l, so it is a "forward weighted

shift".

Combining Lemma 2.7 and Theorem A, we easily obtain the following:

C O R O L L A R Y 2.8.

*C (i) Each Hm is a reducing subspace for C'C, so C*C = ~ m ~ z C IH.~.

(ii) For m > 1,Hm is spanned by the eigenfunctions {gm_l,n}n~=l of C*C; for m >_ O,H.~ is

spanned bu the eigenfunctions {f-m,nL%l"


(vi)

N o t e : Observe that

(iii) For each m 6 Z, C : H,~ -* H,~-I.

1 (iv) The singular numbers of CIH,~ satisfy s~ ~ ml)-g-]~, n = 1,2, . . - .

(v) CIH.~ C $1,~, S~.~ is t he smallest ideal containing CIHm and

(i.e., those norms are bounded below and above by positive constants independent of m).

(p_~ ) ,/v 1 F o r p > l , l l C l . . , l l . ~ . ~ . ~ , (1/p+ l/q= O.

L~I-IH,~ = Cz " ~ , m > 0 ;

D , n H ~ = C S m , m < 0 ;

L~NH_,~ = / ~ 2 ~ O H m = { 0 } ; m > 0 .

w S p e c t r a l E s t i m a t e s of CIL~(D).

Let L~(D) = L~ = (closure of polynomials in L~}. P : L e --+ L~ is the orthogonal

projection. As is well-known (cf. e.g. [B]),

f 1 (P f)(z) = JD f(w) (1 - z~) 2 dA(w).

K(z, w) := (1 - zz~) -2 is the Bergman kernel in D.

(i)

(ii)

(iii)

P R O P O S I T I O N 3.1. Let m = 0, 1 ,2 , . . . . Then, we have:

CZ,~= { 5 ; m = 0

z m - l ( k p - 1) ;m > 1

0 ;m=O pCz m ~ zm-I

m + l , m > l

t

C'Cz "~= ) 1-1z l ;ra=O

[ zm - , ~ + I ] ; m > 1


(i~) 1

PC.Cz m -~ = 2z m rn(rn + 2)

; m = O

; m > l .

Proof:

C(zm) = JD wm dA(w) = L + /~ " z - w I<M I<1~1<1

fM<M 1 _w ~ = 2f~l<l~l 1 w/z dA(w)

= _ ~ E 1 f wm+kdA(w) z k>o 7 JJ~l<l~l

[O ;m > 1

Win-1 f f dA(w)

= --k~>oZkfzl<M<lWm-l-kdA(w)

= ~ 0 ; r n = 0

[ zm-l([z[ 2 - 1 ) ;m _> 1

This proves (i).

( i i ) Clearly, P2 = 0, so PC1 = 0. For m _> 1, we have

PC(z'~)(z) = /D w ' n - I ( I w [ 2 -- 1) dA(w)

= ~ ( k + 1)? i'D ~'-'~k(Iwl~ - l ) d A ( w ) k>O

~0 1 Zm--1

= mz'n-1 tm- ld t - rn + 1

(iii) Note tha t (C'f) = fD ~ d A ( w ) . For m = 0,

fi~ ~ dA(w) = -C (z ) ( , )= ~ -I~1 ~. ( c * c l ) ( z ) = , ~ -

Formm _> 1,


(c*c1)(4 ~ - - /D wra--l-(--]WJ2---- 1)dn(W) = ~wl<Jz,~-~zl<,w[<l 1

_ __1 ~ 2 _ k ft~l<l~lwm_,(iwl2 - 1)@kdA(w )

- tm- l ( t - 1)dt - 1' 2 m JO m m +

while (cf. proof of (ii)) the second integral vanishes.

(iv) For m > 1,

P(C*C~)(~)

F o r m ~ 0~

_ ~ f l _ 1~,1 ~ ' ~

= k~O(k+l)ZkiDWkW'~ m + l J

= (m + 1)z TM t m t dt -- z m. m + 1 m ( m + 2)

P C * C I ( z ) = P(1 - I z l ~)

/o /o 1 = ~ ( k + 1)z ~ w~(t - Iwl2)dA(w) = (1 - I w l ~ d A ( ~ ) = ~. k>O

The Proposition is proved, o

The following simple fact is well-known. We include its proof for the reader's convenience.

L E M M A 3.2. Let H be a Hilbert space and let T : H ~ H be a bounded operator. Let

{em}~=o be an orthogonal system of (non-zero) vectors in H such that

(i) f m = Te .~ ,m = O, 1,2,- . - , are pairwise orthogonal;

(ii) T j ( ~ ) ) ; = O;

(iii) limm~oo Ellmll = O.

Then T is compact and its singular numbers are the non-increasing rearrangement of the

sequence {lifmLI/lle~ll}.

P r o o f : Set u.~ = e.~/llCmll,Vm = fm/llfmH (where if f,~ = O, we define vm = 0). Then,

T ~ E am(%Um)~m m>O


where a m = I l f m l l / i l e m l l , a ~ --+ o. So,

T* = ~-~.am(',Vm)Um; m>O

T*T = ~_, a~(',Um)Um, m>_O

(T'T)'~2 = Y~ a,~(',um)um. rn>0

and

The l a t t e r is the spec t ra l resolut ion of ITI = (T'T) 1/2: the e igenvectors are U m = em/ll~,~ II,

and the e igenvalues are am,m = 0 , 1 , 2 , . - . . Since am -~ 0, IT I is compac t . Hence T is

compac t (since T = VITl-the polar decomposi t ion) and the s ingular number s of T are

precisely the {am} rea r ranged in a non-decreasing order. []


( i) The singular numbers of CP are

so = 1/X/2 , s m - ~/m(m + 2)

f o r m = l , 2 , . . . .

In particular CP C $1,~ and S1,~ is the smallest unitary ideal containing CP.

{ 1 ( i i ) The singular numbers of PCP are . ~ m=l" Thus P C P E $1,~ and $1,~ is the

minimal unitary ideal containing PCP.

( i i i ) The singular numbers of C*CP satisfy Sm ~ -~. So, C*CP E Sil~,~ is the smallest

unity ideal containing C*CP.

P r o o f : Let T = CP, em = z m. Then according to Propos i t ion 3. l ( i )

{ 2 ; m = O fm(z) = (Tem)(Z) =

z~-'(zlzl 2-1) ; m > l

Since Ilfmll ~ = ' r = m(m+~(m+:) ' II~mll ~ = a - ~ ' m _ > 1 a n d I I f~ l l / l l~ml l = ~ ,

(if) Let T = PCP, era(z) = z m,

Z

f m ( Z ) = ( T e m ) ( Z ) = P C ( z ra) ~- z ra-1

roT1

, m = O

, m > l "

2 I I f~ l lV I l~ml l ~ ' Then , ao = 0 and a m := -- (m-V1),~'

this proves (i).


(iii) T = C*CP, e,~ = z TM. Applying Proposition 3.l(iii) we obtain

1 -Lzl 2

Then, ][foil 2 = 1/3, and for m > 1

Thus

; m = O

and the Corollary follows.

; m > _ l

t )2 5 m + 6 d t =

+ I .~(m + i)~(.~ + 2)(.~ + 3)"

1 I 1 s II a m : = ]]Jmll/][eml[

m S

R E M A R K : Comparing Corollaries 3.3 and 2.5, one can say that C P and P C P are "two

times smoother" than C. Also, from Proposition 3.1(iv) it follows that the eigenfunctions

of P C * C P are precisely the monomials z "~, m = 0, 1 , . . . with eigenvalues Ao = 1/2, A,~ = 2 r~(m+2) as m >_ 1. So (of. Theorem A), the operator C P is indeed a much "nicer" operator

than C.

The action of C on L~ (= the conjugate analytic functions in L 2) is even more regular

than its action on L~. Analogues of Proposition 3.1 and Corollary 3.3 can be combined into

the following proposition which we merely state without a proof.

P R O P O S I T I O N 3.4.

(1) c ( ~ " ) = ~"+' ~--Ti-, m >_ O.

(ii) For f C L](D) , C ( f ) ( z ) = f~ f(~)dr

(iii) /5C/5 = CP , i.e., L](D) is an invariant subspace for C. (P : L 2 ~ L ~ is the orthogonal

projection.)

zm (1 (iv) c * c ( ~ ~ ) = a - ~ , - Izl~), '~ > o.

(v) p c * c ( ~ TM) - ' m. >_ o. (m+l~(m+2)Z ~ m

(vi) The singular numbers of C P are x/('~+~)("~+~) m=0" So C P C $1,oo and $1,~ is the

minimal unitary ideal containing CP.

v~ , so C*CP E $1/2,oo and (viii) The singular numbers of C*CP are ('~+l)x/(m+2)(m+a) -~=0

$1/2,~ is the smallest unitary ideal containing C*CP.


R E M A R K : It would be nice to be able to extend qualitative results of this section to

more general simply connected, bounded domains. However, the simple argument given in

w (Corollary 2.6) breaks down for the operators CP, C*CP, etc. By a variational method

one could probably obtain an analogue of Corollary 2.6 for domains obtained by a small

per turbat ion of a disk. Still a complete analogue of Corollary 2.6 seems to require some new

ideas.

w W e i g h t e d E s t i m a t e s .

Consider two radially symmetric probability measures d#~(z) = ~dt~(r2) ,d##(z) =

2~d/~(r2), z = re i~ For x > 0, let

/01 /01 a(x) = t ~ ( t ) ; b(x) = t~d#( t ) .

So, in particular, for m = 0, 1, 2,. �9 �9

a(m) z m 2 . = fl r l~( .o) , b(m) z TM = II ] l - ( . ~ r

Consider the operator C as an operator from L~(#~) into L2(#z), and C P , : L~(#,)

5~(##), where P~: L2(#~) ~ L~(#~) is the orthogonal projection and i~(#~) is the subspace

of analytic functions in L2(#~). Since #~ is radially symmetric, from P roposition 3.1(i) it

follows that {Cz m} are orthogonal in L2(##). Thus, by Lemma 3.2, the singular numbers

of C[L~(,~) : L~(#~) -~ L2(##) are I I C ( = ~ ) l l ~ / l l = ~ l l ~ = IlC(='~)llz/a(~)'/~, rearranged in

non-increasing order.

We have (Proposition 3.1(i)):

/r /: 11c115 = z l2d#p = t d # ( t ) = b ( 1 ) .

F o r m _> 1,

/0' IlCz"llg = ~rn-l(1 -- t)2d#(t) = b(m - 1) - 2b(m) + b(m + 1) = A~b(m)

(A'~f(x) := f ( x + h) + f ( x - h) - 2f(x)) . Therefore, we obtain the loll owing corollary.

2 C O R O L L A R Y 4.1. The singular numbers of CP~ : L=(#o) --+ L2(##) are the non-

increasing rearrangement of f {~b_~_)(~) ~ ~/2~ I.~ a(m) / Jm>o'

Now, take

d#( t ) = / 3 ( l - t ) # - I , / 3 > 0 ;

dc~(t) = c~(l - t )= - Id t , c~ > O.


Note that for a = / 3 = 1,L2(#~) = L2(##) = L~). We have

m ! b(m) = (/3 H- 1)--. (# + m ) '

m! a ( . ~ ) = (~ + 1)... (~ + ~n)"

According to the celebrated Euler-Gauss formula, for x > 0

Hence,

Also, one easily finds that

SO

Thus, we obtain the following.

m~ " ' - -~ (x + 1 ) . . . (x + m) m~

F(x + 1).

1 b(m)

rrt#

1 a(~) ~

1 A~b(.~) ~ m~+~'

( A~b(.~) ) 1/~ 1 a(m) ] "~ m(#-~+2)/:"

C O R O L L A R Y 4.2. Let dc~(t) = a(1 - t)~-ldt, d/3(t) = /3(1 - t)Z-ldt, (a,/3 > 0).

Then,

(i) T,,# := Clni(,o) : L~(# , ) ~ L2(#~) is compact if and only if ~ + 2 > a, In this case, the

singular numbers of T~,z satisfy s,~ ~ m -(~-~+2)/2.

(ii) T~,Z is bounded if and only if fl + 2 > alpha.

2 is the smallest unitary ideal containing T~,Z. (iii) If/3 + 2 > a, then Sp,oo with p - ~ta-~+2

( iv) T~,~ 6 Sp, 0 < p < co, if and only if a < fl + 2_ where ! + ! = 1. In particular, Ta,# is q P q

Hilbert-Schmidt if and only i ra < fl + 1, and it is trace class if and only if a < ft.

( i )

R E M A R K S :

Note that Corollary 4.2 holds even if a or fl = 0. This case corresponds to the Lebesgue

measure on the circle (L2(#o) = L 2 (T~ ~ ) and L~(#o) = H2(T) - the Hardy space ).


(ii) If a = / ~ = 1, both spaces are L2(D) and the singular numbers of T1,1 = C P behave

asymptotically like • So, as we have already seen earlier, the smallest unitary ideal m" containing CP is $1,o~ (Corollary 3.3(i).

(iii) The above discussion can be adjusted to other negative values of a (or, ~), i.e. we

= E,~=0 f ( m ) z could e.g. replace L2,(ff~) by the space H~ of analytic functions f ( z ) o~ ^ ,~

in D such that

Ilfl[~ := ~ [/(m)12( m + 1) -~ < or m>_0

Accordingly, for/3 < 0, consider HZ instead of L2(#Z). Since C maps analytic functions

vanishing at the origin to analytic functions times (1 - ]z[2),

f If(z)l~ Mb'+~dA(z) Ilfll~+~. I l c f l l ~ = ~ JD - - - ~ ( 1 - ~-

(iv)

So, 1

m , 8 + 2 �9

For /3 > - 2 , we can interpret C as the identity operator from Ha into H~+2. The ( 1 "~1/2

singular numbers then behave asymptotically like (ma/m~+2) 1/2 = t,m~---~] and

Corollary 4.2 applies.

The methods of this section can be used in the study of operators o f the form CM~,

where M~g = ~g and ~(z) = f([z[ 2) is a radial function, as an operators from L~2(#a)

into L2(fZ). Indeed, let F(x) = f~f ( t )d t . Then CM~ is defined on the analytic

polynomials if and only if the numbers

{ fo 1 f~t l2 dt~(t) ; n = 0

I[CM~(z~)[]2L~("'~ = flo tn-l[F(1 ) -F(t)J2dl~(t) ; n > O'

are finite. Since {CM~(z~)}~=o are orthogonal in L2(#~), T~,~,~ :-= CM~ : L~(#a) --+

L2(ff~) is bounded if and only if the sequence

w~ = r l C M ~ ( ~ ) l r . ( ~ S I I z ~ l l ~ ( . o l , ~ = 1 , 2 , . . .

T~,~,~ is compact if and only if lim=~oo w~ = 0. In the special case where ~(z) = 7(1 -

Iz lb ",-1 , (~, > o), d#,.(,-r ~) = o~(1 - . -~ '~' - l -"-~' j ,,~ ~ and dff~(,-r ~'~) = ~(1 - , '~ ) . -1 d(,'~)~,d" it is easy to see that w~ ~ n -('+(~-~)/2). Thus one obtains the genera l iza t ion of

Corollary 4.2 (which corresponds to the case 7 = l) with p-1 = 7 + (fl - a)/2 and

T~,z,~o in place of To,~. The case 7 = 0 corresponds to the Cauchy operator on T.


w Spectral Est imates of the Logari thmic Potential .

Let f~ C C be a bounded domain. For f 6 L2(f~) define

(Lf ) (z ) = f(w)log

It is clear that L is compact (even Hilbert-Schmidt). In this section we shal 1 study spectral

estimates for L similar to those attained in w for C. Let us start with some simple obs

ervations

L E M M A 5.1. Let f~ = D. For all 3~ E T(= 0D) L commutes with the rotations R~,

i.e. R~L = LR~. ((R~f)(z) = f(Az).)

C O R O L L A R Y 5.2. {L(z~)}~=o are orthogonal in L2(D).

Proof: As R~ is an isometry of L2(D),

(n(z~), L(zm)) = (R~L(z~), R~L(z'~)) = (LR~(z'~), LR~(z~)) = A~-~(L(z~), n(z'~))

for all A e T. Hence, (L(z"), L(z'~)) = 0 if n # m. []

L E M M A 5.3. Let z = pe ;~ 6 D. Then

e~'~~176 _re~0l ~ = ~,., . ~ ~ ,~ ; . > 0

Proofi If r < p, then

e i'~~ log ~ - rei~ ' --2~r

fo [ ~ l ( 1 ( 1 ) ) ] d O = 2~ei~~ log + ~ log (l_~ei(0_O) +log 1--~ei('-~ 2---~

fo 2~ ( ~ ) dO l~(r /p )k fo2~e ik ( t_o , einodO = e i ~ ~ log ~ + 2 k 2~r

logl/p ; n = 0

2n , n > O

For r > p, a similar calculation yields

2, e~01og( 1 ) d 0 = ) ' l ~

~. 2n \T/ ; n > 0


L E M M A 5.4. For n > 2, pn+2 (i) L(z")(pe it) = ~'"-~5 [ 2 - ~ + �89 - p2)];

(ii) JlL(zn)][ ~ 1/nS/2;

(iii) HL(zn)[I/][zNI ~ 1/n 2.

P r o o f : By Lemm a 5.3 we have

cin t [ / P ' 2n+l j[pl ] C i n t [ p n+2 L ( z ' ) ( P ~ ; ~ ) = po p~ dr + p>~r = - -

n n (2n + 2)

rIenCe,

_ _ 1 ,~ _ / ) ] + ~p (1 j

IIL(zn)][ 2 1{ /ol = - - - - [ 2n+3(1 p~)dp n 2 P2n(1 - P2)22pdp + 2n + 2 .Io p " -

} -+ (2n + 2) ~ P2'~+SdP

- n 2 - n + l n + 2 n - ~ + k

} (n + 1)(2n + 4)(2n + 6)

So Lz~z~l ~ n r ~ 1 n512 ~ n~. []

C O R O L L A R Y 5.5. Let P : L2(D) --, L~(D) be, as above, the Bergman projection.

Then the singular numbers of LP satisfy s~ .~ ~ . Hence, LP E $1/2,oo and $1/2,~ is the smallest unitary ideal containing LP.

L E M M A 5.6. For n > 2,

(i) F L ( z n) = z~ n(~-+2) '

(i i) ItPL(zn)ll/ltz=lt = ~(~-+2),1. J

(iii) sn(PLP) ~ ~ .

P r o o f : As

P f ( z ) f (w) dA(w) = /D ( l ~ w z ) 2

= y~.(k + 1)z ~ fD f (w)~kdd(w) ' k>0


we can obtain from Lemma 5.4(i)

l f2~ei~~ [ r~+2 1 ~ ] PL(z ~) = F~(k + 1)~ fo ~o ~ - [ ~ T ~ + ~r (1 - r ~) ~ e - ~ ~ k>0 2~r

] = n ~- r 2n+l dr = n(n + 2) n z~ L~+-~ + - r~+3

This is (i). (ii) is trivial and (iii) follows from (ii) and Lemma 3.2. []

C O R O L L A R Y 5.7. P L P E $1/2,~, and $1/2,~ is the smallest unitary ideal containing

P L P .

R E M A R K : Thus, (cf. Corollary 3.3) both operators PC*CP, P L P have diagonal

form with respect to the basis {z ~} of L~ and their spectra are boundedly equivalent. This

is by no means an accidental (cf. also [AKL, w Remark (ii)]). It seems promising to study

the relationship between these two operators further, in particular for domains other than

D.

Let us say a few words concerning the "size" of L on L 2 = L2(D)). T he following result

was obtained in [AKL, Theorem 3.1].

T H E O R E M B. Let ~ = D. The eigenvalues of L : L ~ --~ L 2 are precisely the numbers 2._2 . ~,~,~ = 3 . . . . rn = 0 ,1 , . - . ; n = 1, .-- (recall that y,~,~ denote the n-th positive zero of

All ~ . . . . m > 1 are double eigenvalues and the corresponding the m-th Bessel function).

eigenspace is spanned by

and

r io gin,n( e ) = Jm,n(jm,nr)e i(m+l)O

while all s are triple eigenvaIues~ and the corresponding eigenspa ces are spanned by

hn(re ie) = J0(j0,~r);

g0,~(r~ ~) = J l ( j0s )~~

fo,~(r~ ~~ = J~(jo,~r)e -~~

Note that positivity of the eigenvalues imply that L is positive definite. Th is is, of

course, well-known but is usually proved by applying the Fourier t ransform (cf. [L, Ch. I]).

Since jm,~ ~ (m + n) (cf. Lemma 2.4) an argument similar to that used in Corollary 2.5

yields the followin g.



(1) The singular numbers of L satisf~ s~(n) ~ ~ n "

(ii) L E &,~o and Si ,~ is the minimal unitary ideal containing L.

(iii) L E &,co for any domain ~ C D.

(iii) follows by the "majorization" argument as in w (Corollary 2.6).

As in w consider the eigenspaces H~ of the rotation group: Hm := { f E L2(D) :

f(eiOz) = e~'~~ V0, Vz}, m = 0, 4-1, 4-2, �9 - ..

The following is an unalogue of Corollary 2.8.

C O R O L L A R Y 5.9. Each of the Hm 's is an invariant (hence, reducing) subspace of L.

Moreover,

2&<_1,. ~ 1/([.~1 +n)Lm r 0, n > 1

and

s,~(LIH0) ~-. 1/j~,., n > 1.

Hence, [[LIH,~H1/2,o o ~ 1, ][Li.~]]oo ~ ~ .

(Here, I] I[~ is the operator norm, i.e. the largest singular value .)

Corollaries 2.8 and 5.9 can be extended without difficulty to a more abstract set t ing.

P R O P O S I T I O N 5.19. Let {Tm}~=l be bounded linear operators on a Hilbert space H

which are pairwise orthogonal, i.e. (range T,~)_l_ (range T~) and(range T~)_l_ (range T;),

for m ~ n. Suppose that for some p : 0 < p < oo

1 HTmlIp,~ < 1, IITmll~ <

_ _ t u l i p �9

Then the operator T = ~ m Tm belongs to S2p,~, and I1TH2p,oo < 21/2p.

P r o o f : We have sn(Tm) < rain { 1 1 } < 2 lip -- nr/p, ,~r/p -- (m+~)I/p" Clearly, the singular numbers of T

are the non- increasing rearrangement of {a~(Tm); m, n _> 1}. Fix t > 0. We have

[{n: s~(T) > t}l = I { (m, ,n) : sn(Tm) > t}l { 21/~ }

-< (~' ~); (~ + ~),/~ > t

= { ( .~ ,n) : ,~ + n _< [ ~ ] }

~ - 1 2 2 - t2p


So,

and IITtt2p,~ ~ 2 a/2p.

~l/2p s~(T) <_ n - i~ , n = 1, 2,.

R E M A R K S :

(i) Similarly to w most of the results in this section admit extensi on to spaces L2(#o),

where d#~(re i~ = c~(1 - r2)~-12rdr~. We omit the details.

(ii) Once again (cf. the discussion in w we see that operator P L P is "two times nicer" than

L itself. Unfortunately, we cannot claim that we really understand this phenomenon

and suggest it here as an interesting topic for further investigation.

R e f e r e n c e s

[AKL] Anderson, J.M.; Khavinson, D. and Lomonosov, V. Spectral properties of some operators arising in potential theory, 1990, preprint, 29pp.

[B]

[GK]

Bergman, S. The Kernel Function and Conformal Mapping, 2rid Ed. Mathematical Surveys, No. 5. Providence, RI; Amer. Math. Soc. (1970).

Gohberg, I.Ts. and Krein, M.G. Introduction to the Theory of Linear Non-Self Adjoint Operators, Amer. Math. Soc., Transl. of Math. Monographs; Providence, Rhode Island; 1969.

ILl

[a]

[s]

[w]

Landkoff, N.S. Foundation of Modern Potential Theory, "Nauks Moscow, 1966 (Russian).

Russo, B. On the Hausdorff-Young theorem for integral operators Pac. J. Math., 68 (1977), pp. 241-253.

Simon, B. Trace Ideals and their Applications, Cambridge University Press, 1979.

Watson, G.N. A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1922.


Jonathan Arazy

Department of Mathematics

University of Haifa

Haifa 31999, ISRAEL

and


University of California

Irvine, CA 92717, U.S.A.

AMS Classification Numbers: 47B10, 46E22

and Dimitry Khavinson


University of Arkansas

Fayetteville, Arkansas 72701, U.S.A.

Submitted: July 23, 1991 Revised: January 7, 1992

Spectral estimates of Cauchy's transform inL 2(?)

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Transcript of Spectral estimates of Cauchy's transform inL 2(?)