Integr. equ. oper. theory 37 (2) 169-207 0378-620X//020169-39 $1.50+0.20/0 �9 Birkt~user Verlag, Basel, 2000

Integral Equations and Operator Theory

R E L A T I O N S B E T W E E N C O N V O L U T I O N T Y P E O P E R A T O R S O N I N T E R V A L S A N D O N T H E H A L F - L I N E *

L.P. CASTRO AND F.-O. SPECK

Dedicated to Ant6nio Ferreira dos Santos on the occasion of his 60th birthday.

This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces, H ~,p, which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is tha t it provides explicit operator ma- trix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of pro- perties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders: s - l i p r 7Z. For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented.

1 INTRODUCTION AND FORMULATION OF THE MAIN THEOREM

The topic of investigation is an operator of the form

14/~,o = raAl~.p(o ) : Hr'V(~) -+ HS'V(~) (1.1)

acting between Bessel potential spaces where r, s E JR, p Ell, co[, ~ is a finite union of intervals, A denotes a translation invariant operator from Hr,v(]R) into Hs'P(]R) and ra the restriction of distributions from S'(IR) to ~. More precisely we have

= ] - co, bo[U]a,, b , [ U . . . U]am, bm[U]am+,, +co[ (1.2)

A ~ = / C , ~ = 5r- l (bA.jr~, ~ E S (1.3)

H r'p = Hr'P(]I~) = A- rL p = J:-IA-~'JrLP(]R) (1.4)

= c H ,P: supp c (1.5)

H"P(f~) = r a Y "'p. (1.6)

*This research was supported by Junta Nacional de Investigaq~m Cientffica e TecnolSgica (Portugal) and the Bundesminister fiir Forschung und Technologie (Germany) within the project Singular Operators - new features and applications, and by a PRAXIS XXI project under the title Factorization of Operators and Applications to Mathematical Physics.

170 Castro, Speck

Here m is a positive integer, - c ~ < b0 < al < bl < " " < am < bm < am+l < + ~ , where the first and the last interval may, or may not, appear. S = S(]R) denotes the Schwartz space of rapidly decreasing smooth functions, $ ' the dual space of generalized functions of slow growth, ]~. a convolution by ]r E $1, 5v the Fourier transformation, ~A E L~oc(lR) the Fourier symbol of .A = ~ . and .k(~) = (~2 + 1)1/2 for ~ E JR. H r,p is equipped with the norm induced by /2, H~,P(~) with the subspace topology and Hs,P(~) with the quotient space topology, respectively. Multi-indexed spaces (r, s C ]R ~) can be considered by analogy as well as scales of Sobolev-Slobodebkii spaces.

Several papers are devoted to the study of the Fredholm property, index formulas and invertibility of ~/Y~A,n, see for instance the work of M. P. Ganin [20], B. V. Pal 'cev [41, 42], V. Yu. Novokshenov [40], Yu. I. Karlovich, I. M. Spitkovsky [23]-[26], M. A. Bastos, A. F. dos Santos [3]-[5], M. A. Bastos, A. F. dos Santos, R. Duduchava [6] and L. P. Castro, F.-O. Speck [14]. While the early work of M. P. Ganin is written in a classical form focusing explicit solution in certain special cases and reduction to Riemann boundary value problems on IR, most of the recent work is based on the construction of an algebraic equivalence after extension relation, see [28, 29] and [6] as well,

o zy = .E o 2-~ F (1.7)

with a matrix Wiener-Hopf operator We = VVr according to the previous case where f~ -- IR+ --]0, oc[, i.e. to find, beside of We, additional Banach spaces ]I, Z and invertible linear operators E, F acting between dense subspaces of the corresponding direct topological sums such that (1.7) holds. If E and F are homeomorphisms, (1.7) is said to be an equivalence after extension relation and we write

~,r L We,. (1.8)

This is equivalent to the fact that Wr and We are matricially coupled and We is an indicator for ]&r see [1, 2].

Both relations (1.7) and (1.8) are reflexive, symmetric and transitive. But evidently (1.8) has much stronger transfer properties:

�9 Wr belongs to the same regularity class (invertibility, Fredholm property, generalized invertibility, normal solvability etc.) as We does, since the operators have isomorphic kernels and images; indices and defect numbers are the same;

�9 explicit formulas for generalized inverses or regularizers of We imply corresponding formulas for those of 1/Yea,a, and vice versa;

�9 qualitative properties of solutions can be concluded, dependent on the particular form of E and F: singular behavior, asymptotic expansion etc. [12, 43, 44];

�9 operator theoretical conclusions are possible: description of the spectrum, numerical range, reduction of order, perturbation, positivity, application of the fixed point prin- ciple, normalization [38] etc.

Castro, Speck 171

The following diagram shows the strongest

f~

]0,a[ u?=daj, bj[ ]0, a[U]b, +oo[ ] - oo, b0[u?=l]aj, +oo[

relations that are presently known

r = s = 0 r, s C IR

*,L algebraic Z * algebraic *

Wea,a E F 4, We C F * We~,a E F 4, We E F

(1, 1) can be easily derived, for instance, from [27] and (3, 1) was proved in [9]. (2, 1) and (4, 1) can be constructed with the methods of [13], but are not presented anywhere. All the equivalence after extension relations constructed with the extension methods of [13] are characterized by operator matrix faetorizations (1.7) having triangular invertible bounded operator matrices E and F with the identity operator on the main diagonal. This fact simplifies considerably certain dependencies between the related operators. That is the case of formulas for (generalized) inverses of the related operators, see [13].

The algebraic equivalence after extension relation (1, 2) was proved in [28] (for a special class of Fourier symbols); such a result was also presented in [29] but for all Fourier symbols that guarantee that We~,a will be a well-defined and bounded linear operator. The authors of [6] use a different definition of equivalent operators (Fredholm property plus coincidence of defect numbers) but prove also the algebraic relation for a wider class of symbols. (2, 2) was proved in [10] (also for Sobolev-Slobode~kii spaces). This was done by using the Kuijper extension method. Also in [4], an algebraic equivalence after extension was proved by the Kuijper method, but only for ~2 =]0, a[U]aql, aq2[, where 0 < a < aql < aq~ and ql, q2 are rational numbers.

The algebraic equivalence after extension relation constructed by the use of the extension method of Kuijper gives invertible linear operators E and F that depend on an algebraic decomposition of the Fourier symbol ~X of We~,a. This decomposition is construc- ted with the help of certain injective and surjective operators which are determined by the geometry of fL Such operators can not be constructed if f~ has semi-infinite intervals. That is the reason why the method can not be applied to the cases (3,2) and (4,2).

The relations in (3, 2) and (4, 2) mean that the two operators We~,a and We are Fredholm operators only at the same time and, in this case, their corresponding defect numbers coincide. Such a result was presented in [26]. Moreover, if one of the operators has the Fredholm property then it is possible to construct an equivalence after extension relation between the operators We~,a and We. This was done in [14] where the result was proved for fl --]0, a[U]b, +c~[ and arbitrary orders of Bessel potential spaces (and also for Sobolev-Slobodehkii spaces).

Previous results than those shown in the diagram can be found, for instance, in [3, 5, 20, 23, 24, 25, 40, 41, 42]. In the case of r, s 7 ~ 0, a useful reasoning in some of these investigations is the lifting [51] procedure, which will also be considered here in the context of intervals.

The main result of this paper is to present a method to construct equivalence after extension relations for arbitrary unions of intervals and orders k = r, s E IR which are not critical, i.e. k - l i P ~ 2Z.

172 Castro, Speck

T H E O R E M 1.1 For any finite union of intervals (1.2) and non critical orders r, s ~ ~ + 1/!9 the convolution type operator We~,n introduced in (1.1) is equivalent after extension to a Wiener-Hopf matrix operator

We A = rR+ AI[Lp+(~)]~ (1.9)

where A = .T-I~A �9 yz ~A E [LO~(IR)] nx~. All formulas are explicit: the operator matrices E and F, which characterize the relation, are algebraic compositions of known convolution operators, Wiener-Hopf operators and projectors; ~A is a matrix function on IR and an algebraic composition of the given q~A~ shift symbols and Ai, where A• = ~ 4- i.

The proof of this result will be presented at the end of w where new tools are already available. Also explicit formulas can be found there, which show the form of ~SA and demonstrate tha t the relation does not depend on a regulari ty proper ty of ,4 or t4;e~,a in contrast to [14, 28].

2 C O N V E N I E N T P R O J E C T O R S , D E C O M P O S I T I O N S A N D E X T E N S I O N O P E R A T O R S

Now we shall consider p Ell , +co[ and non critical space orders k = r, s, i.e. k - 1/!9 E IR \ 2Z. For these orders, we denote by k' = k'(p, k) the integers so tha t

k ' - 1 + 1/p < k < k ' + l / p . (2.1)

Consequently, k = k~(p, k )+ k"(p, k) = k '+ k", where the la t ter number is zero or a fractional o n e .

L E M M A 2.1 P• = Xa• E E (H~'P(IR)) if and only if s E] - 1 + 1/!9, l/p[.

P R O O F : From [45], w Theorem 1, proposition (i), we know the above charac- terization for the characteristic function Xa+ as pointwise multiplier in H~'P(IR). The proof for Xn_ follows by reflection. []

D E F I N I T I O N 2.1 On Hs,P(IR), let us define the operators

= A+ P+A+, P_~'P = A - P_A_, (2.2) I ~ _ P s t s t s t s I = A - P+A_, II~ p = A+ P_A+. (2.3)

where AS ~' = ~- lA• ~ = + �9 , A• ~4-i, and P• : H~",P(IR) -+ H~",P(IR) are the complemen- tary projectors onto the subspaces of distributions supported on the positive and negative half-line, respectively.

Note that 1-I~:P = P~_'P = P+ and 1-IS_ '; = p_~,v = p_ for s E] - 1 + 1/p, 1/p[ which are bounded.

P R O P O S I T I O N 2.2 The operators P~='P and IIfr p fulfill the identities

P~-'P + II~ p = IH~,~(m (2.4)

P2 p + n$ ~ = I.~,.(~3 (2.5) i m P S " = H~'P(IR+) = kerI I~ p (2.6)

i m P _ ~'p = Hs'v(]t{_) = kerI I~ p. (2.7)

Castro, Speck 173

P R O O F : The first two identities, (2.4) and (2.5), are directly checked by the use of Definition 2.1. The other two propositions are a consequence of the definition of Hs'P(1R) spaces

~o e H*'P(IR) iff A~:~ e LP(]R) (2.8)

and of the fact tha t A_g*' are bijections from ~rs-r177 to Hs,P(IR• e.g.

pf,Pgs,'(]R) = ASP• = AZfP•162

= A_7:S'HS-8"'(IR• = ~rs'p(IR• (2.9)

considering ~ to be an element of Hs'P(IR), then the following assertions are equivalent:

n : ~ = o (2.1o) --81 8 ! h~: P-4-A:Fqo = 0 (2.11)

8 t P.A:r = 0 (2.12)

st ~p-r ~ (2.13) A:F~ E w~:FJ

(p e Hs'P(]R@. (2.14)

[ ]

P R O P O S I T I O N 2.3 l:)s,P I-rsvp s

P R O O F : The result is a consequence of Lemma 2.1 and of the isomorphic character of A~ 8. More explicitly, for the non critical fractional orders, one obtains

--81 81 8 ! IIPs176 ---- IIA• P•177177177 8 t -< I IA*~ I IH. - . , , , ( , , )= I1~o11,,..,(~), (2.15)

--81 8 t ~ 81 = --A=-~D , , = IIA:~ P+&~IIH,,,(R) II ~ H - * , . (~ )

8 ! < I I&~l l , , . - . , , . (~)= I1<1.~,.(~). (2.16) [ ]

Therefore, the operators presented in Definition 2.1 are projectors acting on Hs,P(]R). We like to describe the images of YI~ p and, for s > - 1 + 1/p, decompose those into Hs,P(IP~) and finite dimensional spaces. For s < - 1 + 1/p the situation will be different, as we shall see.

D E F I N I T I O N 2.2 Let s > - 1 + 1/p. We shall use the subspace of Hs,P(IR) defined by

{0}, if s ' = O H•'P (IR) = (2.17)

l ~ + _ + R s p a n { ~ / j : j E ~ , j < s ' } , if s'_> 1

where 'Tj(x) = xJ- le-X,x > 0, and l~P+_a~ : H*,P(IR+) --+ Hs,P(]R) is a continuous extension operator (which always exists due to the possible smooth extensions of xJ-le -x, x > O, onto the full line).

174 Castro, Speck

holds.

L E M M A 2.4 If s > - l + l / p , then the decomposition into a direct topological sum

H~'P(]R-) @ HsjP(]R) �9 H~'P(]R+) = Hs'P(]R~) (2.18)

P R O O F : See Theorem 3.1 in [49]. []

C O R O L L A R Y 2.5 If s > - 1 + 1/p, then the following diagram for the images of the projectors P~:'P and II~ p is valid:

H~"(a-) �9

P :'~ H;~,, ( a )

P R O O F : The result Lemma 2.4.

^

H:,~(~) �9 H~"(a+) = H~"(~). (2.19)

appears as a consequence of Proposition 2.2 and []

loH~"(a• (2.24)

[]

= H~'P(]R), (2.22)

= H ~ " ( a ) , (2.23)

H~,P(]R)).

P R O O F : Noticing that

H~,'(~_) + H~"(~+)

g~"(a_) n H~"(~+)

then the isomorphic identifications

H~,P(N)/HS,P(]R~=) ~_

yield the statement.

D E F I N I T I O N 2.3 If s < - 1 + l /p, we define H~'P(]R) as the subspace of H~,P(]R) whose distributions are concentrated at zero, i.e.

{ } H~"(a) = ~ e S' (a) : ~ = E c Y -1~, cj e C . (2.20) j= l

L E M M A 2.6 If s < - l + l / p , then the decomposition into a direct topological sum

loHS'P(]R_) @ H2'P(]R ) @ IoHS'P(]R+) = Hs'P(IR) (2.21)

holds (where lo denotes the operator of extension by zero from the correspondent space to

Castro, Speck 175

C O R O L L A R Y 2.7 Let s < -1 + 1/p. The following diagram for the images of the projectors P~'P and II~ ~ is valid:

P~'~H~,'(~) rI~fH',~(~) ^ ~,

loH~'V(lR-) �9 H~'P(IR) @ log"P(]R+) = H~,~(~) (2.25)

P R O O F : See Proposit ion 2.2 and Lemma 2.6. []

R E M A R K 2.8 The corresponding extension operators are given by

I~{P+_~ = II~:vI : Hs 'P(~+) --+ Hs'P(]R) (2.26)

where I is any (element-wise) extension and the operator does not depend on the choice of I.

T H E O R E M 2.9 For every union of intervals

r~ =] - oo, b0[u]a,, b~[u.., u]a,.,,, ~,,,[u]~,,,+,, +oo[ (~.zz)

projectors P onto Hr'P(~2) and II along Hs'P(~ ') can be represented in the form

P P~ - - r t r ! X]~i,+oo[A+ A ~_~ (A+ ~' A '''~ A -r ' A r' - " ~' = = - - , - - X]b,+oo[ - ) + + X]~+l,+oo[ + + A _ X]-oo,b0[A_, .?'=1

(2.28) 8 t 8 ! _ - - - - 8 / 8 !

j = l (2.29)

where the last two elements in the sums on the right-hand sides of (2.28) and (2.29) appear if and only if the corresponding last and first intervals in the unions of (2.27) do exist.

P R O O F : As notat ion we use, for c > 0, X]+c,+~[" = T+cP+T~:c, X]-~r177 = T• , where T• = U - I ~ . • U (with ~-~(~) = e ~ , ~ E JR) are the right and left shift operators, respectively. The result is a reorganization of the projectors przp and H~ p due to intervals different from JR• For tha t it is only necessary to use compositions of those projectors with convenient shift operators. []

C O R O L L A R Y 2.10 P and II are higher order convolution type operators with oscillating symbols in Bessel potential spaces, more precisely they belong to the algebra gene- rated as follows

P, ~ e alg {P• T~, i~:: c e ~ , k ~ ~} . (2.3O)

176 Castro, Speck

3 E Q U I V A L E N C E A F T E R E X T E N S I O N W I T H A P A I R E D O P E R A T O R AND L I F T I N G

We like to generalize the Gohberg-Shinbrot formula [21, 47]

P,,41px * P A P + Q = (I - P A Q ) ( P A + Q) (3.1)

to an asymmetric space setting

HAIPx * H A P + P'BH' (3.2)

where P'BIn,X is an invertible operator between the complemented subspaces

r I ' x = (I - p ) x , p ' Y = (I - H)Y. (3.3)

This relation (equivalence after extension by an invertible operator) is not more general than equivalence after extension since (e.g.)

[ H A I p x 0 ] [ r I A i p x O J [ I l p x 0 J (3.4) 0 P'Biri, z = 0 IIp,y 0 P'BIn, x

makes sense (but, in general, this is not useful because of the inconvenient space setting). However, we ask not only for any invertible operator P'BIH,x explicitly, but for one

of special form: a Wiener-Hopf operator of higher order.

L E M M A 3.1 Let s > - 1 + 1 / p . Then A--r maps the following (complemented) subspaces bijectively

Hs'P(]R-) @ Ha'P(]R) G H"P(]R+) = H"P(]R)

^ ^

HS"(a_) �9 Ha,'(a) , HS,'(a+) = H~,"(a)

and the inverse operators are given by the corresponding restrictions of A+*'A r

P R O O F : From Proposition 2.2 and Lemma 2.4, we know that

Hs'V(IR~-)

Hs'~(~• �9 H2'(~)

Therefore, _S 1 81

AT~*'A~H*,P(IR•

(3.5)

= A T-CA• ~' k(A-*'P• ~A*'HS'P(r~ v=~U = A2:*'PTH*"'v(IR)

= A;eP~A~H*'V(Ia) = _gr*'P(a:F), (3.8)

= aT-s' A~" (a~" e ~ A f w , , ( ~ ) ) = A; 8, e:~u s",p(a)

= A~:8'P~A~:H"'P(IR) = ~rs'P(IR+) �9 H~'v(IR), (3.9)

= P:~'VHS'V(IR), (3.6)

= H~PHS'P(IR). (3.7)

Castro, Speck 177

L E M M A 3.2 Let s < -1 + 1/p. Then A-~'A~ maps the following (complemented) subspaces bijectively

IoH"P(]R-) G H~'v(]R) �9 loH"P(]l~+) = H~'P(]R)

$ A-~'A~ $ $

lom,~(~_) �9 H; ' (~) �9 loH~'~(~+) = H~'~(~)

(3.1o)

and the inverse operators are given by the corresponding restrictions of A+~'A~.

P R O O F : The result is proved by the same reasoning as in the proof of Lemma 3.1, using here Proposition 2.2 and Lemma 2.6. []

Therefore, using the modified projectors (for the asymmetric space setting) and an algorithm for the construction of equivalence relations between higher order Wiener-Hopf operators [13], the desired explicit equivalence after extension relation appears. For that it is crucial to write the invertible operator P'BI~,X in terms of a higher order Wiener-Hopf operator. Here we shall present such constructions for two representative cases: a finite interval and a union of a finite and a semi-infinite interval. All the other cases can be derived by the same arguments. Therefore the proof of the general situation, i.e. Theorem 1.1, is presented at the end of the section.

Recall that in all situations we suppose that the orders of the spaces are not critical (cf. the beginning of w

3.1 The Fin i te In terval We shall denote A_/),+ by ~. Recall that LP(]R) -- P+/2(IR) = H~ and

~(~) = P ~ , ~ ~ JR.

T H E O R E M 3.3 For ~ =]0, a[, we have

~Ym~,~ * Wr C E ([L~_(]R)] 2) (3.11)

where

(by = ~ + ~ ~

We split the proof of Theorem 3.3 into 6 steps which are presented in the following propositions.

P R O P O S I T I O N 3.4 For ~ =]0, a[, ~ = ]R\-~, a homeomorphism

/~: II~,PH"V(]R) -+ P~;PH~'P(]R) (3.13)

8~p from alg {A~:, P• T~ : k, c E JR} is given by B = P~, Bin~TH,.p(~ ) = BIn~,pH,.p(~ ) where

r / l _ s r I r l l _ s . B = A_ P_A+ +A+ X]~,+or A" (3.14)

178 Castro, Speck

and the multiplication operator can be written as X]~,+~[' = T~P+T_~. Moreover

kerB = P[fPH~,P(]R) = Hr'p(f~), i m B = P~lPHS'P(IR) = Hs,P(IT). (3.15)

P R O O F : From Lemma 3.1, Lemma 3.2 and the mapping properties of the projec- tors P~'P, II~: p (see Corollary 2.5 and Corollary 2.7) we get a homeomorphism

T Ar-S(A-r ~ (3.16)

which can be written in a simpler form. Moreover, considering ~ to be an element of H r'p (IR), the following assertions are equivalent (see Proposition 2.2):

B ~ -= 0 (3.17)

A ~ - 8 ( A : r ' A ~ ) n ~ ~ a T - ~ ( a ; ~ ' A ~ ) n ; ~ r _ o ~ _ �9 o + = 0 (3.18)

{ E 2 ~ = o (3.19) IZgT_~ = 0

{ - �9 Hr'P(]R+) (3.20) T_o~ �9 H~,P(~_)

e Hr'P(f2). (3.21)

In addition, using Proposition 2.2, corollaries 2.5 and 2.7, lemmas 3.1 and 3.2, one has

BHr,P(IR) : Ar--sp:'PHr,'(]R) + TaA~+-sP;"T_aH~"(IR)

= A[-SH~,P(IR_) + TaA~-:H"P(IR+)

= Hs,P(]R_) + TaHS'P(a+)

= g*'P(a'). (3.22)

[]

R E M A R K 3.5 Due to the nature of the present projectors, Proposition 3.4 is also valid for critical orders of s, i.e. it is valid for arbitrary s E JR, if we consider the image space Hs'P(f~ ') in (3.13).

P R O P O S I T I O N 3.6 The convolution type operator We~,a (see (1.1)) on f~ = ]0,a[ satisfies the relation

" F ~ , f ~ * Vrs 'PA Dr'P IDs'P lrtVrr'P

~ r / r t

= ( A : 8 , . § _ ~+_8, ~ o , + ~ �9 ~ ) A (A+ . § - ~_-~' ~o,+~. ~ ) + A~'-sP-A:; + A 5 - % o , +~ �9 a ~'_ (3.23)

= r .

Castro, Speck 179

PROOF: The relation is obtained as in (3.2)-(3.4). The final form of the operator denoted by T and presented in (3.23) is derived by using the projectors generally identified in Theorem 2.9 and the invertible operator B of Proposition 3.4. []

P R O P O S I T I O N 3.7 The convolution type operator ]/Y~a,a on f~ =]0, a[ is equi- valent after extension to a paired convolution type operator of higher (algebraic) order, i.e.

To e alg {2p-1r $-, Ak,, p__: r e L~(IR), k e IR}, (3.24)

acting from Hr"'P(IR) to Hr The relation does not depend on the particular form of A.

PROOF: We lift the last operator T (see (3.23)) so that (cf. Proposition 3.6)

w~,a ~* a ~ : r a j ' = To (3.25)

which has the form

~"~ = Ar162 = To: H ' (]R) --+ Hr To P+Ao + P+Ao + P-Ar P- (3.26)

where

A0 : AS,(,_ -s, A< A"aS) A+ X]a,+oo[" +) A (A+ P+ - A -r' - - X]a,+~[" -

+(A,_'A+r r" s" A+ - X]a,+~[" A~A-7 e. (3.27)

[]

P R O P O S I T I O N 3.8 The convolution type operator Wv.~,a defined in (1.1), with ft =]0, a[, is equivalent after extension to

T(Ao) : H~"'P(]R+) --+ ~rs"'P(lR+), T(Ao) = P+Aqp+HT,,,p(R ), (3.28)

with Ao given in (3.27).

PROOF: If we choose

II = P+: Hs"'P(IR) -+ Hr (3.29) P = P+: Hr"'P(1R) --+ Hr"'P(]R), (3.30) II' = P_ : H~",P(IR) --+ Hr"'P(~), (3.31) P' = P _ : gs"'P(~) -~ He"P(~) (3.32)

(where these are bounded operators, due to the fact that -1 + l ip < r", s" < 1/p), then the result is a consequence of (3.2)-(3.4) and (3.25). []

Let us denote by ~ the "entirely lifted" Fourier symbol of .4 so that

8 ! r I = ; ~ A u . (3.33)

180 Castro, Speck

P R O P O S I T I O N 3.9 We have

diag [P+Aolff,,, p(R+), I~,",p(R+)' I~,",~(R+)'/k,",~(r~+)] = w,_owr o

W~ We,, W~o = E1 0 0

0 0

where E1 and F1 are bounded invertible operators.

0 0 0 0

I~,,,~(rt+ ) 0 0 I~,,,,,(L )

F1 (3.34)

PROOF: From (3.27) one has

T(Ao) = P+AoI~,,,(R+)

: . ro_ (w~_~,owq (. ,~_owr ( .~ , . , .q (w~_ow,_.,o) + (.,,.,w.o)(w,_:_.,.)(.,._o.~,~,) + (w,., . , .o). , ~,,_.,, (.,~_o.,r ,~+ (3.35)

Therefore, applying the method for the construction of equivalence after extension relations between higher order and pure Wiener-Hopf operators presented in [13] and using some algebraic combinations, we obtain formula (3.34) where E1 and F1 are defined by

E1 : Y"",.(r~+) �9 Y'",P(~+) �9 Y'" , " (~+) �9 Y'",p(~+) ---+ Y'","(P~+) �9 Y'",p(~{+) �9 ~'",~(R+) �9 .Y~",P(~+)

E1

(E1)n -Iff,".~(a+)

= 0

0

(E1)n 0 0 0 I~,,,,,(~+) 0

W~_o Wr -we_,,_,,+ - w ~,,_,,, - I ~ , , <r~+ ) A+ 0 0 I~,,,.~(x{+ )

(3.36)

/ \

w i t h (~,),, = -wr + wr ~ wr + w~:;,_,,,) and (S.).~ = I~.,,,r Wr162 ~

Y'",p(}t+) �9 Y'",p(~{+) �9 Y~",p(~{+) �9 Y'",~(a{+) -~ -~'",~(~{+ ) �9 Y~"'p(~+ ) �9 ~'",~(~{+ ) �9 ~"'~(~+ )

I~r,.,,,p (rt+) 0 0 0

w~_~ wr z~,,<~+~ o o 0 0 0 Ifi,,,.p(~{+)

(3.37)

with 1~12, = ( ~ . . . . . '~ + ~ : ' ,") ~ o~,~' - ~ o ~ "o []

Castro, Speck 181

R E M A R K 3.10 The identity (3.34) presents an explicit equivalence after exten- sion relation between T(Ao) = P+AolP/,p(~+) : ~rr"'P(]R+) -+ ~rs"'P(]R+) and

w+~ ~ : _y+',~(~+) + ~+',,(~+) __+ ~r+',,(~+) + Y+',~(r~+)

W+uo = p+~:-l[ ~-o C~+~A+ ~' 7o~ ].3:.r (3.3s)

Finally, we are now in a position to relate Wr in the sense of equivalence (after extension), to an operator acting between/2 spaces. For that we have to lift the operator Wvuo , defined in (3.38), to the 12 spaces.

P R O P O S I T I O N 3.11 The operator Wcvo , presented in (3.38), is equivalent to

W+t~ : [L~(IR)]2_+ [L~_(IR)]2 ' W + v = p+.T._l [ 7._ a Cr 0 J ),,+~),u ~-o 4, �9 7 . (3.39)

P R O O F : Note that

Wr = E2 Wv~o F2 (3.40)

where E2 and F2 are bounded invertible operators defined by

E2: ~rr @ ~rs"'P(]R+) --+ [L~_(]R)] 2, E2 = p+~--1 diag [~ ' , A!']..T', (3.41)

Fz: [n~_(~)] 2 -+ ~rr @ Hg"P(]R+), F2 = p+~--1 diag [A+ ~'', A+s"] �9 U. (3.42)

[]

Let us recall the relations constructed in this subsection in the following chain of relations (where ~ and * denote equivalence and equivalence after extension relations between bounded linear operators, respectively)

W+a,n * T ~ To * T ( Ao ) L Wvv ~ ~ Wov (3.43)

due to (1) extension to a paired operator, (2) entire lifting on fractional order spaces, (3) the generalized Gohberg-Shinbrot formula, (4) reduction of higher order Wiener-Hopf operators, and (5) lifting on (zero order) /2 spaces.

Due to the transitivity of ~ and * , we obtain an (explicit) equivalence after exten- sion relation between the operators W..~,n and Wr Therefore, the statement of Theorem 3.3 is proved. "

3.2 The Union of a Fini te and a Semi-Infinite Interval

T H E O R E M 3.12 Let f2 =]0, a[U]b, +cx~[. We have

V~+~,n * W+ v E s ([L~_(]I~)] 6) (3.44)

182 Castro, Speck

where

~ V

,~5~t;~7" ~*r= a*_~,~7"rb ~""rb -A~_~aSg'r=r - ' ' 0 O 0 0 -C*-2*'rb_~ 0 _~e'

--~r"r-a 0 ~r"%_a 0 0 ~r" 0 ~*%-b 0 2~ ~'' 0 0

-~'r_~ 0 0 0 ~" 0 0 0 0 0 0 -('"r~-b

�9 (3.45)

We split the proof of the above theorem into 6 steps which are presented in the following propositions�9

P R O P O S I T I O N 3.13 For f2 =]0, a[U]b, +oo[, f~' = IR\-~, a homeomorphism

J~: I-I~Hr'p(]R) --+ P~lPH~"(IR) (3.46)

s,p from alg {A~, P+, T~ : k, c E IR} is given by B = P~, BIH~;H,,,(a ) = Blng;H,,p(a ) where

r It - -8 __Td .gl B = A~'-*P_A~ + (A+ X]~,+oo[' A~) (TbA_ P-A+T-b) (3.47)

and X]=,+~o[" = T~P+T_~. Moreover

kerB = H"V(g~), imB = H*'P(ft'). (3.48)

PROOF: From lemmas 3.1 and 3.2 and the mapping properties of the projec- tors P~'P and II~ p (see Corollary 2.5 and Corollary 2.7) we get a homeomorphism B ----

Bfri~rHr,p(R), with

B = A2-~ (A2r A~_) H2" + (T~A~+-S (A+" A~) H~_PT_a) (Tb (A-" A~) FI2"T_b) (3.49)

which can be written in the form of (3.47). Moreover, the inverse of/~ is explicitly given by

-1 = r ' r ' r ' s r " (3.50) Slpa;,H,.,(~t) (A+r'p_AS_-r"+TbA+ P_a_Ta_ba- P+A; 7 T-a)lpa;,H,.,(R ).

The proof of (3.48) is done in a similar way as in the proof of Proposition 3.4 and therefore is omitted here. []

P R O P O S I T I O N 3.14 The convolution type operator )/Vr (see (1.1)) on l't = ]0, a[ o >, +oo[ satisfie, the relation

W c a , ~ * ITs ,PA l9 r,p A_ ps,PI317r~P

.a = ( A : - < : + A_

- - r ' r ' - - - - r ' A~) �9 (A+ P+A+ A_-")q~,+~[ �9 A ~" + A+ X>,+or ~l.lt_ 8 l,I r l l _ s _ r I ?'!

---- T .

(3.51)

Castro, Speck 183

P R O O F : See (3.2), the projectors identified in Theorem 2.9 and the invertible operator B defined in Proposition 3.13. []

P R O P O S I T I O N 3.15 Let t~ = ]0, a[ U ]b,+oo[. The convolution type operator ]/Y+~,fl is equivalent after extension to a paired convolution type operator of higher (algebraic) order,

To e alg {~L--Ir . .~ly, Ak,, P=t=: (~ e n~176 k e JR}, (3.52)

acting between Bessel potential spaces of orders k E] - 1 + 1/p, 1/p[. The relation does not depend on the particular form of A.

P R O O F : We lift the last operator T in such a way that

W * ACTA+ ~' To (3.53)

with To defined by

To: Hr"'P(]R) ---+ Hr To = P+Ao + A~"-8" P- = P+Ao + P-A/_'-r P- (3.54)

where

A0 A" ( I - -" �9 A:; -" �9 A f ) . 4 _ A+ X]~,+oo[ + A_ X>,+~[

�9 (A+r'p+ - A-r'x]a,+oo[. AfA+ r' + A+r'x>,+oo[ ")

(3.55)

[]

P R O P O S I T I O N 3.16 The convolution type operator W+A,a defined in (1.1), with - ]0, a[ U ]b, +oo[, is equivalent after extension to

T(Ao) : ~r~"'P(]R+) --+ H 'P(]R+), T(Ao) = P+AolP+H:,,.p(R ), (3.56)

with Ao given in (3.55).

P R O O F : The same reasoning as in Proposition 3.8. []

P R O P O S I T I O N 3.17 The operator

T(Ao) : H:""(]R+) -+ H "(]R+), T(Ao) = P+Aq~,,,.,(~+),

defined in (3.56), is equivalent after extension to

W + o : Yr" , ' (a+) + Y '" , ' ( a+) + Yr","(~+) + Y'",~(~+) + [H+"~(~§ +

W+:o = P+~-~+Vo �9 .,~

(3.57)

[Y'",. (3.58)

184 Castro, Speck

where

g2vo =

,I~ (*'r~ rbv~ rb - ~ r ~ : - " 0 _$1 t i t__S11

0 0 0 -ff rb-~ 0 -A+ -r-a 0 %-a 0 0 1

0 %-b~*' 0 2 0 0 - r _ ~ r' 0 '0 0 1 0

0 0 0 0 0 -%-b

(3.59)

f?4 : F4 =

so that

W~ v = E4 W~vo F4. (3.66)

P R O O F : Using (3.55) we are able to write

T(Ao) = P+AoL~>,, p(rt+ )

= W r - W~r162 + W~bW~_ b

+we,~o (w~-~,-.,~w~_oe, - WT_o~-~,~ - w~_.,~o_oW~_b

+w~o (w~_o~ + w~w~_b - w._~_~,~W~_o~,)

+ w , ~,,_~,, (w~_o- w~b ow~ ~) (3.60)

Then we apply the methods of [13] to (3.60) which leads us to the equivalence after extension relation

[ P + A ~ W ~ v ~ ] 0 Iz . _ = Ea 0 i[ff,,,p(~+)]~ F3 (3.61)

where t t 4

and Ea, F3 are bounded invertible linear operators. Due to the size of such operators we avoid to present them here. We refer to [11] for the definitions of these operators. []

P R O P O S I T I O N 3.18 The operator Wcvo, presented in Proposition 3.17, is equi- valent to

W+ v : [L~(]R)] 6 --+ [L~_(]R)] 6, W+ v = P+.,~-I(P V . a ~ (3.63)

where +y is defined in (3.~5).

P R O O F : We construct invertible bounded operators

- - ~ r t l 8 rl r t t r t! E4 = P+.~'-ldiag [.~a_",/~," A_,/~_,/~_,A_] ..~', (3.64)

[LP (]R)] 6 -+ H""'P(]R+) �9 Hs""(]R+) ~3/ff"'~(]R+) @ H~"'P(]R+) @ [H"'P(]R+)] 2

p+j~-I diag [A+'", r "" -~ . . . . " - ' " . ;~+ , ~+ , A+ , ~+ ,;,+ ] ~ (3.65)

Castro, Speck 185

R E M A R K 3.19 If [ . 0 0] W ~ : [L~_(]P~)] 3 --+ [L~_(]I~)] 3, W ~ -- P+5 r -1 0 Ta-br 8 0 �9 .~

Air ~ -T_~r ~ ~!r

is a Fredholm operator, then from Theorem 3.5 in [14] we know that ]4]r n (with [2 = ]0, a[ O

]b, +oe D is equivalent after extension to W~. This fact demonstrates that, at least in some cases, the matrix ~v defined in (3.45) is not of minimal size among all symbols of Wiener- Hopf operators that act be tween /2 spaces on the positive half-line and are equivalent after extension to ]W~,fl-

For the special case of r = s = 0, other operators equivalent after extension with ]a]v~,a (for ~ = ]0, a[ U ]b, § are constructed in [9].

3.3 P r o o f o f T h e o r e m 1.1

P R O O F : Let [2 be an arbitrary union of intervals, like (1.2). Once more, consider ~' : ]R\~ and, similarly to Theorem 2.9, construct projectors P~P and II[~, p.

Therefore, we are able to get a homeomorphism

/~ s,p II~,PH~'P(IR) -+ P~PHS'P(]R) (3.67) : _P~, BIII~PH~,p(R) :

by choosing appropriate elements from alg {Ak,, P• T~ : k, c E JR}. This is always possible and is strongly dependent on the geometry of ~, see (3.14) and (3.47).

Using (3.4) and the homeomorphism/3, one obtains an equivalence after extension relation between YYr and

rr~,PAD~,P p~,Pnrr~,P (3.68) T= ~ r § ~_,~,.

Moreover, this implies that

]42~,a L To = A~TA+ r' (3.69)

where To is a paired convolution type operator of higher order from alg {9v-1r �9 $-, Ak,, P+ : r E L~(]R), k E ]R}, and so that the elements 9r-1r v are composed only with P+ (and not with P_). This particular characteristic of To allows us to consider the projectors (3.29)- (3.32) in the relations (3.2)-(3.4) in such a way that (see Proposition 3.8 and Proposi- tion 3.16)

To * We o (3.70)

where We o is a general Wiener-Hopf operator acting from Hr"'P(IR+) to Hs",P(IR+). Now, rewriting Wr as a higher order Wiener-Hopf operator, we are able to apply

to Wr the iteration method presented in [13]. This leads us to an equivalence after exten- sion relation between Wr and a new Wiener-Hopf operator matrix acting between Bessel potential operators with orders k E] - 1 + l /p, l/p[. Then, using Bessel potential operators, we lift the latter operator o n t o / 2 spaces. Following this reasoning one obtains

W~o * W~ A (3.71)

186 Castro, Speck

where We A is a Wiener-Hopf operator acting between [L~(]R)] ~ spaces, for some n �9 IN. Finally, using the transitivity property of the equivalence after extension relation,

we have

]A)~,a ~ W~ a (3.72)

where the relation is explicitly determined because, following the above reasoning, all the indicated relations-are given in explicit form. []

4 S O M E A P P L I C A T I O N S

In this section we like to present some results, which cannot be concluded from the algebraic equivalence after extension relation or from equivalence of the Fredholm property.

In particular, we will consider the case of g2 =]0, a[ and different classes of Fourier symbols 0.4 where we can effectively present (generalized) inverses of Wr This will be done following the general reasoning that if we have an equivalence after extension relation (1.8), the knowledge of a (generalized) inverse W~- of the operator Wv allows the construction of a (generalized) inverse of 142r in the form

W~,~,a=R(F-I[ Wg 0 0 Iz] E-l) (4.1)

where E, F are invertible bounded linear operators given by the equivalence after extension relation and R denotes the restriction to the first component spaces (see [15]).

4.1 O n A l m o s t P e r i o d i c Four i e r S y m b o l s

In the first application let us focus on convolution type operators acting between Bessel potential spaces, with the same integer order of smoothness and on a finite interval, i.e., let us consider 1/Vr defined in (1.1) with r = s c 2~ and f2 =]0, a[. Moreover, assume that the Fourier symbol of the operator is of the following almost periodic type

q~A(~) = -b + b ~ bke i"~ (r 0). (4.2) k = t

This means that ~5 A belongs to the smallest closed subalgebra of L~(]R) that contains the singletons %(#) = e i'~, # �9 IR. This subalgebra is usually denoted by AP. For • �9 AP the Bohr mean-value is the number

M(r = lim 1 f_~ t-+oo~ tr (4.3)

We say that r �9 AP ~• admits a (right) AP-factorization (see [25]) if it can be represented in the form

r = ~_ diag [%1, . . . , %,] 9+ (4.4)

where the real numbers #j are ordered decreasingly and r E AP[ • r E AP~. • with

AP+ = {r E AP: M(7_, r = 0, for 4- # < 0}. (4.5)

Castro, Speck 187

The numbers #j do not depend on the choice of the representation (4.4) and are called the partial AP-indices of %b. Moreover, (4.4) is said to be a (right) APP-faetorization of r if

r E [APP_] nxn = l A P p (-1 AP_] nxn (4.6)

where AP p denotes the smallest closed subalgebra of t h e / 2 Fourier multipliers that contains T., # e IR.

L E M M A 4.1 [25] Let ~ fulfills (~.2). If the Fourier spectrum of (~.4, {# e ]R : M(T-,~A) ~ 0}, is contained in ] - a, a[, and either 0 > ~'1 >"" > ,m or 0 < "1 <"" < "m holds, then

~(~) = [ T_~A T~0] (4.8)

is canonically (right) APP-factorable by

~(~) = @(~_) ~(0) (4.9) u+

where

7._ a OL+T_a [ ~2A fl+ ] if O>~'l > ' ' ' > Pm

�9 (~_) = (4 .1o ) [-~ 1 0 if O< ul < " - < ~'m

I 1 - a + [0 1 ] if O > u l > ' " > ~ m

(I)(~ (4.11) u+ =

[ (PA ~-a ] if O < ~ l < ' ' ' < l J m ~ _ & - T a

with

a• E b-lcoT:~a+o "" fl• ~ E co bk "r:~a+~'k+o'", (4.12) mEN• k=l •ENk,•

e" u is the inner product of the vectors Q = (QI,..., Om) and u = (ul .. . . , urn),

co = (~i + .... + ~m)!~o~O~ o~ QI!Q2!'"~m! ~I ~2 ""bm,

N~ = {~ = ( ~ 1 , . . . , era) : ~ 1 , . . . , Om S 2Z+, 7:~" ~ < a},

Nk,a: is the set of vectors Q E IV. such that increasing Ok by 1 takes e out of N•

(4.13)

(4.14)

188 Castro, Speck

PROOF: See Theorem 2.4 in [25] where a left canonical APV-factorization of ~(u ~ is obtained. []

As an immediate consequence of Lemma 4.1, we have the following lemma.

L E M M A 4.2 If the conditions of Lemma 4.1 are fulfilled, then we can construct the following well-defined and bounded linear operator

W-I p+f-1 k(O(~ ~ P + ' ~ - L ( ~ ) ) - L ' ~ = [ " _ (W_~(xo) . . . . (W:(lo)~ \(W-~(z~ ] 12 (4.15)

where (W2o~) , (W:L5 (W:~o~) , (W5~o,5 a~ethe components (acting betweenS~(a) \ #U .]ii \ CU ,]12' k ~U -]21 \ ~>u ,]22

spaces) of W-~(lo) which are determined by (4.10) and (4.11).

L E M M A 4.3 Let r = s = k E 2~ and ~ =]0, a[. The following (explicit) equiva- lence after extension relation holds

0 l,~k,~(a,)

where F

E5 = ] r~tII~'VHk'v(~) [ 0

= E s [ P+A~ Ip_L,(R)O ] Fs, (4.16)

0 ] A- k I~k.p(a,) _ (IL,(~t) + P+AoP-) (4.17)

~ l U-1 ' (4.18)

Ao and B are defined in (3.27) and (3.13), respectively.

PROOF: Observing that rflliik, VHk,p(l~) : II~'PHk'p(]P~) --+ Hk'P(~) is invertible by

II~'Vl, where l : Hk,P(12) --+ Hk,v(]R) is any extension operator, it follows that E5 and F5 are invertible bounded linear operators. Therefore, the result appears by a combination of propositions 3.4, 3.6 and 3.7. []

L E M M A 4.4 Let r = s = k E 2~ and ~ =]0, a[. The following {explicit) equiva- lence after extension relation holds

[ W~,~ 0 0 I~,,(~,) 0 0

0] 0 0 : Es O I[L~_ (]r

/[Lv(R)] a 0 0 of 0 Fs,

Ip_L~(n)

(4.19)

Castro, Speck 189

where

W ~ ) : [L~_(IR)] ~ --+ [LP(]R)] ~,

W ~ ) ,~ ":'-'-~(~) m(~) ~-_~ r 0

and Es, F~ are invertible bounded linear operators defined by

(4.20)

= O 0 /[L~(I~)]~ 0

0 0 ] [ j E6 0 Ip_Lp(R ) E1 0

J~[Lp (]l~)] 3 0 0 ~P_Lp(~B.) '

= O 0 IlL 0 I[~(~)1~ O IP_L~(R) 0 Ip_L:~(R ) % R)]s

with E~, F1, E~ and F~ presented in (3.'35), (3.37), (4.17) and (4.1S), respectively.

(4.21)

(4.22)

P R O O F : From Proposition 3.9, we already have (3.34) in the case of r = s = k E ~ , i.e., (3.34) with r ' = s' = k and r" = s" = 0. Therefore (4.19) follows from the application of the transitive property of the equivalence after extension relation to (4.16) and (3.34), for the above parameters. []

T H E O R E M 4.5 Let ~ be of the form of (~.2). I f the Fourier spectrum of ~.~, ( # E ] R : M(f_~(I)A)#0}, is contained in ] - a,a[, and either 0 > v~ > .. . > u,~ or 0 < vl < .. . <Um holds, then

Wr : ~rk,p(~) ~ Hk,p(f~), Wr = r . . ~ - a r

with ~ =]0, a[ and k E 2Z (p C]I, c~[), is generalized invertible by

(f l ]) , ~ 0 0

W ~ , a = R F~ 1 0 I[L~_(~)]3 0 E~ -1 , (4.23)

0 0 Ip_Lp(R)

where E6 and F6 are defined in (4.21) and (4.22), respectively. Moreover, if k = 0 then Wc.~,n is a (two-sided) invertible operator and (4.23) is the inverse of W~z,n which (in this ease) takes the form

W ~ n = (W~(lo)) W~.~ + (W-~Zo,~ + (W~-(lo)) Wr (4.24) ' \ Cv / 1 1 - \ Ou -'/12 \ wv / 1 2

P R O O F : For the easiest case k = 0, due to the properties of the relations es- tablished in w (cf. also (4.1) and (4.19)), we have that W~A, ~ is an invertible operator. Moreover, such relations enable us to construct the inverse of W~,a . As a mat ter of fact,

190 Castro, Speck

considering the inverses of E, and F~ (see (3.36) and (3.37) for r = s = 0), the inverse of 14]~,a is given by (cf. (4.1))

- i % (I,:,,o(R) W~a,a = R 0 Ip_Lv(P~)

[ W'-I R 0

cAo

\ % 111

R , (W2o,) \ %12z

0

0

Ip_Lp(~ ) 0

r } 12 (W:?o,)

% 122 0

-P+Aolp-Lp(P~) ] ) Ip_Lp(I~.)

~ ~ / 0 0 E{-1

IL~(a) 0 0 tc~_(a)

\ %111 - \ %112 \ % 112

For the other cases (k ~ 0), we begin by observing that W~) , see (4.20), is equi- valent to a corresponding operator acting between Bessel potential spaces with order of smoothness equal to k. In fact,

W~(v~,) = (P+9 ~-z diag[Ak_, Ak_] �9 .T'l k';) Wr (P+.T -~ diag [A+'~, ),+k] �9 .T), (4.26)

with

: . . . . +~ - u "5c, (4.27)

and where l k'p : Hk'V(lR+ --+ Hk'P(IR is a continuous extension operator. Instead of studying directly W~) , we consider the restricted/(continuously) ex-

tended operator

l0 Rst 14]~) : [L~_(IR)] 2 -+ [L~(]R)] 2, k < 0

lo Ext 14)v~, : [L~(IR)] 2 + [L~(IR)] 2, k > 0

I 0 W ~ ? : [L~(~)] 2 -~ [L~(a) ] 2, k = 0

(4.28)

From Lemma 4.1 and Lemma 4.2, we obtain that W~) is an invertible operator with inverse

W-I r defined in (4.15). Therefore, the knowledge of the inverse of Wr and the Shift

Theorem [8, 19], lead us to the following generalized inverse of W~):

14;~) = (~'-ldiag[A+k,A+k]. 5 r) I/V~(-~, (P+~--tdiag[A~,Ak_] �9 5el<P). (4.29)

Thus, (4.23) follows from formula (4.1) and from the equivalence (after extension) reIations (4.19) and (4.26). []

Cas~o, Speck 191

R E M A R K 4.6 It is clear that the method of this subsection can be applied to other almost periodic function classes (different from that of Lemma 4.1). For that, beside the relations established in the present work, we need to have an APP-factorization of the corresponding (I)(u ~ This is the case of [7], where an explicit APP-factorization of ~(u ~ is effectively constructed (for a particular class of (I)A C A P ) . Therefore, under the conditions of [7] (see (6.1), (6.2) and Theorem 6.1 in [7]), we obtain a correspondent Theorem 4.5. That is, we obtain an (explicit) generalized inverse VP~ n given by (4.23), with W~-(~o) obtained

from this new canonical APP-factorization (as in Lemma 4.2).

4.2 On W i e n e r Four i e r Symbo l s (~A

Let us now consider operator (1.1) when ~A is an invertible element in the Wiener algebra on the real line

r e ~ W(IR) (4.30)

and

=]0, a[, p = 2, ~ = , = k e 2z. (4.31)

A Wiener-Hopf factorization of ~u is well-known in the case of ~ , = 1 and k = 0, because

which implies a formula for the inverse of We~,a in the case of a sectoriat symbol

(~.A 7-- 1 -~-{ , Ng[tLOO(]R,) < 1 (4.33)

in terms of a Neumann series (provided k = 0 and p = 2 are satisfied). In the present subsection we present a procedure to obtain a generalized inverse of

(1.1) under the assumptions (4.30)-(4.31) for general k E ~ and some restrictions, more precisely the formulas are:

(a) explicit in closed analytical form, if ~A is rational;

(b) explicit in analytical form plus Neumann series, if ~A is not rational.

The strategy is as follows (i) Letting

w = wind (I)a, (4.34)

we consider, instead of

the restricted/(continuously) extended operator

Wr I Rst r a F - I ( P A �9 .7" ,n = Ext r n ~ - - l ~ - $-

( where 8 = - w .

H ' , ~ ( ~ ) -~ H' ,~( f i ) , s > k H' ,~ (~ ) -~ H ' , ~ ( ~ ) , s < k H' ,~ (n ) + H ' , ~ ( n ) , ~ = k

(4.35)

192 Castrol Speck

(ii) We relate the operator (4.35) in the sense of Theorem 3.3 with a Wiener-Hopf operator

,n ~ ) = l + J ~ v " ~ ' : [ L IR)] 2 2

__ o 1 ~AC ~ ~oC ~

where wind (r = 0. This is explicitly presented in (4.19), taking s in the place of k.

(iii) Now we consider the particular cases

o r

~ A ~ = ~ 0 r + if s > 0

�9 ~ C s = ~ 0 r _ if s < 0

where r satisfies (4.33) and r+ E G T~-~(IR), i.e., r~. are invertible rational functions

which are restrictions to IR = IR td {oc} of holomorphic functions in the upper/lower half-plane and continuous in the union of the upper/lower half-plane with the real line.

(iv) So we have to factor (after elementary factorization), cf. [20],

C = ~o T~(~r- -1 = ~o T~p2

where either r_ = 1 (for s > 0) or r+ = 1 (for s < 0), and all non-oscillating symbols are 1 at c~

r = r+(oo) = r _ ( o c ) = pl(c~) = p2(oc) = 1.

(v) We reduce G to a non-oscillating symbol by using (4.32)

G = oo 1 0 r - ~ ( P l - ~ o ) P l - ~ o + P 2 0 - 1

1 0 0 - i '

(vi) Consider the principal part of Go

1 ~%(1 - P2) ] G1 = T-a(pl --1) f l l - - l + p 2

in the two cases s > 0 or s < 0, respectively, separately:

Castro, Speck 193

(vi+) s _> 0, pl = C r ~ 1, p2 = C

a~ = T_~(C~; ~ -1 )

= , -_o (Cr;_ 1 - 1)

%(1 - C s) ] Car+ 1 - I + C a ]

(2sr +l 0 1

1%(1 - C a) ]

A

where b_+b+~ 2a = r_~(~%+1-1), i.e., b_ = P_ r_~(~%+1-1) and b+ = P+ r_~(~-ar+ 1- ~-2s);

(vi_) s < 0, Pl = C a, P2 ---- C at-1

[1 a l = r - ~ ( C - 1 ) [1

= ~ - - o ( C - 1)

[ 1 o] l, 1 C - I + C r -~ = r-a(C--1) 1 0 C2ar -I

c_ i c+ 1[o where c+ + c_( 2a = %(1 - ~%-_1), i.e., c+ = P+%(1 - ( S r -1) and c_ = p _ % ( f - e s _ C-~:~).

(vii) Thus we obtain a one-sided inverse of Wr in both cases and ind W~;) = - 2 s = 2w.

Therefore, from (ii) we have a one-sided inverse of W(~)n . Consequently, similarly with what we did in the proof of Theorem 4.5, we obtain a generalized inverse of Wv.~,n, for the present case.

4.3 A p p l i c a t i o n s in D i f f r a c t i o n T h e o r y

We consider boundary value and/or transmission problems in weak formulation which originate from diffraction of t ime-harmonic waves by an infinite strip, see [22, 32, 34, 36, 39, 52] for background. Let

E = {x = (xl,xe) Z IR2 :x l e [0, a],x2 = 0} = [0, a] x {0} (4.36)

r ' = (lR\]0, a D x {0}. (4.37)

We look for u e LP(IR 2) such that

?.t ::t=

L u ~ =

[u]~, =

where p E]l,oe[, l > 1/p,

gj(Xl),

Umxr~ • E Hz"(IR x IR+)

( A + k 2 ) u • in ] l t x ] R •

( ~ + ( x ) - ~ - ( x ) ) j = ~ , : o

Ou au-

I - 1/!9 ~ I'q, k E C, with ~(k) > 0 are given and

+ ~ + b;, , jD~'u-(xl, O) E,n+r u (Xl, O) +

Xl E [0, a], j = 1,2

(4.38)

(4.39)

(4.40)

(4.41)

(4.42)

194 Castro, Speck

where ~r = (~ , , a2 ) ,m = (mi,m2) E ]N~, b~j E r and gj E H'@-~J'P(]O,a[) are assumed to be known.

R E M A R K 4.7 1. From the physical point of view, one is mainly interested in solutions in the energy space, u E HI'2(IR 2 \ E) [34]. But we know already from the study of half-plane problems that many of these are ill-posed in that space setting [35, 36, 50]. They need a normalization which is often implemented by change of the space parameters 1,p of H ~,p. Another reason to consider the problem in a scale of spaces is to look for regularity results and asymptotic expansion [44]. I I . Boundary values are taken in the sense of the trace theorem. The choice of the data spaces results from the representation formula (4.48) as a consequence of (4.40), which also allows to drop the assumption l > 1/p [17]. The orders of the boundary operators Bj are arbitrary [35, 46, 50]. I I I . We associate an operator with the problem, say

p = p(l,p) : u ~ g = (gl, g2) (4.43)

where the domain �9 of P is characterized by (4.38)-(4.41), the action and the image space of P are described by (4.42) with the corresponding norms. Evidently the problem (4.36)-(4.42) is well-posed in this space setting if and only if the operator

2 l-- i__mj ,p P : X ~ y , X = HI'P(]R x lit+) x Hz'P(]R x JR_), y = x j= iH ~ (]0, a D (4.44)

is bounded invertible. If necessary, we shall point out the space parameters in P(~,P) and other terms in order to avoid confusion. IV. The main objectives are (i) to find the spaces in which the operator p(l,p) is bounded invertible and those where it is normally solvable (which implies the Fredholm property in the elliptic case and the existence of a generalized inverse in terms of factorization); (ii) to determine the defect numbers (not only the index) of P by computing the partial indices of a matrix symbol, which is not only algebraically but also topologically related; (iii) to get a generalized inverse of Pq'P), if possible (a) in closed analytical form or (b) in terms of a uniformly convergent series under physically reasonable assumptions on the parameters. As a mat ter of fact it is not possible to deduce these results only from an algebraic equivalence after extension relation between P and a Wiener-Hopf operator, cf. [6].

P R O P O S I T I O N 4.8 The operator P is (algebraically and topologically) equiva- lent to a convolution type operator on the interval [0, a] in the corresponding boundary data spaces of Bessel potentials. More precisely

P = W~x,a B_ To (4.45)

where the trace operator

l i 2 To : :D(P) --+ [H-~'P(]R)] , TOu = uo (Uo, u o ) = ( u ~ : o , u~=0) t (4.46)

Castro, Speck 195

is bounded invertible by the representation formula

u = ]Cu0 (4.47)

U(Xx,X2) = .Ti2xl {e-t{{)z25+({) X+(X2) + et({)~5o({) X-(X2)}. (4.48)

Here ]C is called Poisson operator, Jr denotes the one-dimensional Fourier transformation, the Fourier transform of ~, X:L the characteristic/unction of ]R• and t({) = ({2 _ k2)1/2

with vertical branch cut across infinity and t({) ~ { at +co. Further

B_ = 5 ~-1 i -I . ~c: L H - ; ' (IR)] -+ Xo = Ht-~ 'v(lR) x HI-~-I'P(IR) (4.49) - t - t

+ _ t maps the Dirichlet trace vector uo = (uo, Uo ) = To u (for u E ~9(P)) into the jump vector of Dirichlet and Neumann data

/ L Oz2 J ~u~, I

(i.e., the single and double layer potentials). The operator ]/Yr : 2(0 -+ Y is an operator of the f o r m ( i . i ) where r = (l - 1 / ; , l - 1 - l / p ) , s = ( l - m l - l / p , 1 - "~2 - i / p ) and

rfo,_<m b§ (bo+2 b + b;,=) ( - i 0 ~ ( - t ( ~ ) ) ~ 2 j ~ j ~

(4.51)

P R O O F : This is an evident extension of considerat ions for the case p = 2, l = 1 and half-plane screens [34, 36, 46]. []

C O R O L L A R Y 4.9 The system of equations

W ~ , n f = g (4.52)

decoupIes, i.e. ~t is triangular after multiplication with a constant matrix, if and only if a linear combination of the two boundary conditions (~.~2) contains only a linear combination of either "difference" or "sum data", i.e. it can be written as

Bu(xl) = ~ b~D'~(u+ =ku-)(xl,0), xl e [O,a]. (4.53) Iol<_mo

So far we did not make use of the fact tha t ~ = [0, a] is a very par t icular subset of ]R which becomes impor tan t now.

T H E O R E M 4 .10 Let ~ be the operator defined by (~.36)-(~.~) and l - 2/p ~ 2Z. Then

P * W~ v 6 C ([L~_(IR)] 2~) (4.54)

196 Castro, Speck

where n = 2 or 1 (in certain decomposing eases described below) and

~ u = [ r_~(~I~ 0 ] (4.55) ) r ~ r~(~I~

where q)A is given by (~.51) or can be replaced by a scalar symbol according to Corollary ~.9 in the case n = 1, respectively. The orders are r = (l - 1/!o, 1 - 1 - 1/p) , s = (l - m~ -

V p , l - ,~2 - l / p ) or , f o r ~ = 1, a r e components of these two vectors.

P R O O F : The case n = 2 is a combination of Theorem 3.3 and Proposition 4.8. In the decomposing case, (b.4 is a triangular (say upper). One can split an operator W~u 2 that has the form (4.54)-(4.55) with n = 1. If it is invertible we have

[w.0 [ , . ][wo 10 wo. 0 ,~ ,456, i.e. equivalence after extension to Wev~ and the remainder operator has the same form (4.54)-(4.55) with n = 1. []

R E M A R K 4.11 I. For technical reasons we extend the definition of A-({) = {:]=i and work now with

~• = { 4- k, D(k) > O. (4.57)

This makes the Fourier symbols simpler since we can combine factors ~ : with t, but does not change the principal nature of factorizations or the topology of Bessel potential spaces Hs,p = A+SL p = A-~L p.

I I . If W~v2 is not invertible, but a shifted one W(:) : H~,P(]R+) --+ Hw,P(]R+) (defined by

restriction or continuous extension) for some w = (Wl,Wl) E ]R 2 one can t ry to consider W(~ 'w) first and then "shift back", i.e. express results for Wr in terms of results for W~(~ '~).

C O R O L L A R Y 4.12 The symbol ~ u can be wri t ten in the f o r m

g2U : Cr [ T-aln 0 ]

where s - r C 77,, ~, i.e. M - ~ and (~-~ are rational, precisely

(4.58)

s - r = ( - m l , l - m 2 ) (4.59)

i f n = 2, which admits integer components up to 1. Moreover, the e lements of iV--~{~4 belong to each of the fol lowing algebras

A = C(fi), C'(fi) (0 < , < 1), Lip(ill, aIg (~l/2, n ( k ) ) . (4.6O)

Castro, Speck 197

R E M A R K 4.13 I. For non-trivial boundary conditions (4.42), the factorization of ~u can never be reduced (by splitting of known plus/minus or rational factors) to the study of a matrix function

G = ~r[ ~-~plb-po ~-ap2b+O I (4.61)

with rational pj and strong plus/minus factors b• which can be factorized explicitly by analytical formulas in the elliptic case, see [20] and Section 4.2. Therefore fixed point argu- ments and sectoriality shall be taken into account. The symbol classes (4.60) are important in the discussion of Fredholm properties, asymptotic behavior, normalization, the multiplier problem and explicit factorization. II. We will now treat a typical example from diffraction theory where the system (4.52) de- couples and one of the occurring symbols in the diagonal of A~_-r ~A is, by fortune, continuous at infinity; but the other one is not, reflecting the typical difficulties.

E X A M P L E 4.14 In the so-called impedance problem the boundary conditions (4.42) have the form

u + + i z + u + = g1 (4.62) u~ - i z - u o = g2

with known impedance numbers z • E (~ due to the two surface materials of the screen. The half-plane case where E = ~ + • {0} (since the dependence of the third variable is dropped) was investigated in various papers [16, 30, 31, 33, 36]. For strip problems in the setting of Bessel potential spaces, there is very little known, see e.g. [6] where only well-posedness is proved for the Rawlin's problem.

From Proposition 4.8 we obtain the Fourier symbol of the convolution type operator )/Pv~,a : H~'V(~) -+ H~'P(~) as

~ = [ - ( t - i z l ) i z 2 1 - i z 2 t - 1 ] - izl t -1 (4.63)

where zl = (z+ + z-) /2, z2 = ( z + - z - ) /2, r = (1-1 /p , l - l - 1 / p ) = (rl, r2), s = ( l - l - 1 / p , l - 1 - 1/p) = (sl, s2) for short.

We treat mainly the case z + = z- , i.e. z2 = 0 and the system (4.52) decouples. Let us denote the two scalar symbols by

~1 = t - izl, ~2 = 1 - iz l t -1. (4.64)

The two corresponding Wiener-Hopf operators (on the half-line) are equivalent to each other in appropriate pairs of spaces (but not in those corresponding to the diffraction problem, see [36]), because

(Ih = A~2(1 - izlt-1)Al+/2, Wr = W~I/~Wr (4.65)

198 Castro, Speck

Thus the factorization of the lifted symbols

59~,o ---- AsJ( t -- i z l ) A + r~ = ~ - ~ - 5 (1 - - iz~t -L) (4 .66)

(b2,o = AL ~ ( 1 - i z l t <) A+ ~ = r189 ~,o

in scales of spaces amounts to the same, see Remark 4.11.I. The situation changes completely, if we consider the interval [0, a] instead of JR+. But before doing this, let us consider the pre-factorizations

l _ ! _ _ 1

@1,o = ~ - ( v ~ + (4.67) ~ . l _ l - - - z

(b2,0 = ~ - ~ v ~+

where ~• = e x p { l ( I 4 -Sn) log~2} E ~W• denote the factors of ~2 in the Wiener algebra since ~2 is a regular element and wind ~2 = 0; so we obtain the following: (j) [ ! : ][ 1 ]

eea, ~ = ~ _ ~ + = ~_A_; ~ o_}_~ ~+A+ z+;+~ o 0 9~_A_ 0 ~+A+ z+}+l (4.68)

represents a "canonical generalized AJP-factorization '' in the sense tha t Wr162 is the

bounded inverse of Wr o in [LP(]R)] 2 if and only i f / C ] } , } + �89 because the exponents of

~_ and ~+1 are in the interval ]} - 1, }[ and those of ~+ and ~-1 in ] - ~, 1 - ~[, see [12, 18];

(jj) the operator ~/V~.~ ~) : Hr,P(~+) --+ H~,v(]R+) is normally solvable (and then Fredholm and one-sided invertible) if and only if

2 ( / - 2/p) • 2Z; (4.69)

(jjj) for all these orders I, one obtains a generalized AAP-factorization of ~ , 0 easily by

~A;(r's) for the modification of (4.68) and thus the defect numbers and one-sided inverses of , ~ a orders mentioned in (4.63) in dependence of I and p.

Let us now consider the operators due to the interval

W%,]0,a[ : Hr~'p(]0, aD -+ HsJ'P(]0, aD, j = 1, 2 (4.70)

i as before and I C]2 2 1 where rl = I - ~, r2 = sl = s2 = l - 1 - v p' p + 2[" Theorem 3.3 implies that

W%,]o,~[ * W~u~ (4.71)

~u~ = ~ - ~ ~--a ~ (4.72) A : ~ ( t - i z l ) ,-~C -1 = { ~ ~ 1- i z~t -~ T~C-~

~ u 2 = ~z-l-}[ 1 -~--~izl t-1 T aO] (4.73)

Castro, Speck 199

which two symbols replace those (4.66) from the half-plane problem (considering still the case where z2 = 0). The last matrix can be factorized as, cf. [20],

1 - i z l t -1 "Ca = 1 0 i z l t - ~ - _ a 1 + i z l t -1 0 --1

with a remainder middle matrix (~2 E ~ W 2x2 since det@2 = 1 and t - l ({ ) = O(l{I -~) as I~f ~ oc. Furthermore wind det (~2 = 0 and @2 admits a canonical faetorization (~ = @2-@2+ in W 2• at least in the case where the (mean) impedance number zl is relatively small, namely if

Iz~l < Kl l t - l l lL~ ; . (4.75)

This is particularly relevant in microwave physics (large Ikl), in acoustics (weakly absorbing surfaces) and electrodynamics (weak electrical conduction on E) [22, 39, 52]. Note that O2 can also be considered as a sectorial matrix function, cf. (4.83). Altogether we have a canonical general ized A P - f a c t o r i z a t i o n in the case 1 e]~, ~ + �89 namely

~ u ~ = 1 0 2- - .Z+ 1~2+ 0 - 1 '

The problem to factorize ~su~ in (4.72) is quite different, of. Remark 4.13, because of the jump at infinity of the additional term ( 1/2. First we use the method of [40] to overcome that difficulty, el. [6]. Let

1 1 ei(Y G(() = 7r--i f-~ - 7 dy (4.77)

which has the properties

where

2 f ~ sin r/ a(~) = ~ Jo - ~ d r ] = sgn { + O(1~1-1) as Ir oo (4.78)

H ~176 for a > 0. (4.79) "r• E •

This allows the following reduction

1 + r ~-aC 1/2 = 1 0 r-a~3 1 + ~P4 0 - 1

r = - i z l t -1 r = cr - ~1/~ + r ~/)3 = ~ - - 1 / 2 - - g - - r ~/)4 = ~--I/2cr + ~1/2Cr -- 1 -- Cr 2 - - r 2 (4.81)

which are all decreasing like CO(l l at infinity and elements of the Wiener algebra [40]. Now we assume that, for some e > 0,

R ( I + r > 2e, j = l , 4 [r -< e, j = 2,3 (4.82)

200 Castro, Speck

which meets again the physically interesting cases of impedance and wave numbers, cf. (4.75), and implies that the middle factor O1 in (4.80) is sectoriah

= ((1 + r + or + + (1 +

_> c( la l [ 2+[a2] 2) for a E C 2. (4.83)

Thus we end up with a canonical generalized SAP-factorization provided l E]}, 2~ + �89

where O1 = O1-O1+ is a canonical factorization in GW 2• Here we like to remark that the case of slightly different impedance numbers where

Iz2[ is sufficiently small in (4.63) can be considered as a perturbat ion of the decomposing case where z2 = 0. This is a consequence of the stability of one-sided inverses against small perturbations. We collect the preceding results in

P R O P O S I T I O N 4.15 I. The impedance problem for the strip (~.36)-(~.g2) with the particular boundary condition (~. 62) is well-posed for arbitrary given p E] 1, oc[, if

1 2 2 1[ (4.85) t c

and (~.75), (~.82) are satisfied provided the difference of the impedance numbers z2 = (z + - z - ) / 2 is sufficiently small. The estimates (~.75), (~.82) do not depend on 1 nor on p.

" 2 + + ~ [ I I . For l E ] ~ + ~, ~ 2 [, J E 2Z, p(l,p) is one-sided invertible under the same conditions and

= - - + = - (4.86) p _ ~ (p(t,p)) if <_ 0

(where intx = max{z E 27,: z < x} and a, ~ denote the defect numbers).

R E M A R K 4.16 The two basic boundary value problems (Dirichlet or Neumann) for strips are - in contrast to the Sommerfeld half-plane case [34]- almost as complicated as the finally treated one due to 59~. Starting with the H 1,2 formulation (p --- 2, l -- 1, see Remark 4.7) we obtain the scalar Fourier symbols, cf. Theorem 4.10, ~A = t -1 and (I)A = t, respectively, so that A : H-1/2'2(IR) -+ H1/2'e(IR) or vice versa in the Dirichlet or Neumann case, respectively.

Now the symbols (I)u corresponding to (4.55) read

1 T~r , (T2uN = 1 Tar -1/2

due to A~2t-IA+ 1/2 = 1, ef. Remark 4.11. But note that the condition 1 - 2 / p ~ 2Z is violated, so that we do not have relation (4.54). Changing p or I or both, we have to factorize (4.87) in

Castro, Speck 201

the first case (p # 2, l = 1) in the sense of a generalized SAP-factorization in L p space and we have to factorize r in the general case (I = 1 + 5, 6 ~ 0) in order to get a generalized

inverse ,,~ �9 D/N" This leads us directly to the questions around (4.80) with r = 0. In the final part we treat the problem of explicit representation of the inverse of

the operator P = p(l,p) associated to the decomposing impedance problem, see (4.36)-(4.44) and (4.62), for admissible parameters (4.85) in terms of the factorizations (4.76), (4.84) -as a "prototype example". We confine ourselves to the case

p = 2, l = 1 + e e l l , 3 /2 [ (4.88)

that seems to be most important in the applications. Other parameter choices like 2 < p < 4, 1 = 1 or 1 < p < 4/3, l = 2 can be treated by analogy; one-sided inverses under the condition (4.69) can be derived by the help of the Shift Theorem [8] or, for critical parameters, by normalization [35, 37].

Here we obtain the following simplifications and abbreviations:

c E ]0, 1/2[, 6 = e - 1/2 El - 1 /2 ,0 [ r = ( e + 1 / 2 , e - 1 / 2 ) = ( 6 + 1 , 5 ) = r ' + r " r ' = (1, 0), T" = (6, 6) 8 = (e - 1 / 2 , ~ - 1 / 2 ) = (6, 6 ) = 8" s' = ( 0 , 0 ) , r " - s" = ( 0 , 0 ) .

(4.89)

Further we suppose that the physical parameters satisfy the conditions that yielded the factorizations (4.76) and (4.84), now with exponents

1 1 1 1 = e, l - - - 1 = 6. (4.90) 2 P P

Therefore the inverse of We v = diag [Wcvl , We,l l is given by the 4 x 4 operator matrix [12, 48]

w;- ) = (4.91)

A• = (4.92)

~+

- 1 - 1 A+ P+A_ f[L~(~t)]4

>--i~>• ,2"

),+~01+ 0 1

0

[ --r'_,~ i ] e1_A ~ 1 0

0

0

A$~02+ 0 1

0

1] e -A -I o

(4.93)

(4.94)

The inverse of 7 ) follows from Proposition 4.8 and the chain of relations presented in Section 3.1, cf. (3.43).

202 Castro, Speck

Step by step we reduce the representation of the inverse of 7 ) to that one of We v in (4.91). First

.p-1

see (4.45)-(4.49). Second

-1

= ](]B_- 1~42;1,fl (4.95)

~r,2~-1 (4.96) = /-'fl I ]imn~/~

where Theorem 2.9 gives us, for r' = (1, O), s' = (0, 0), p = 2,

imrI~ 2 = im (X+- T~x+T-~): [Hs'2(f~)] 2 (4.97)

: 0 IH~.2(I~) X+" 0 IH~,2(p~) -- 0 IH~,2(R) 0 [He,2(p9 (4.98)

which are quite simple (we recall that T~ = ~-~ - :P is the a-shift operator (a > 0)). Proposition 3.6 yields

T : H(~+IJ)'2(]I~) -+ H(sJ)'2(]R)

T = (P+-T~P+T_~)A[ A+IP+A+-A-IT~P+T-~A-o P+ - T~P+T_~O ]

+ [ P_A+ - T~P+T_aA_o P- + TaP+T-~,O ] . (4.99)

The relation (3.25) to the "entirely lifted" operator To is evident

Z 0 = A~TA; r ' = T [ A~-10 In,,2(a)0 ] : H<'2(IR) ~ H~"'2(IR) (4.100)

which is a higher (algebraic) order Wiener-Hopf operator of (pseudo-differential operator) order zero, see (3.24) and (4.89), respectively. (3.27) and (3.28) read

A0 = (I[H~'~(rgl~ - T~P+T-~) A [ A+IP+ - A--1T~P+T-~A-A+IO P+ - T~P+T_~O ]

+T~P+T-a [ A-A+I O IH~,2(R) (4.101)

r(Ao) : P+AoI[~r~,2(R+)]2 : P+Tot[~,2(a+)]2. (4.102)

The next relation, step (4), involves 8 x 8 matrices according to Proposition 3.9:

W._or 0 0 0 W~ W~o 0 0

d iag . .[T(A~ = E1 0 0 I[~.~(~+)]~ 0 F1 (4.103)

0 0 0 I[~6.~(a+)]~

Cas~o, Speck 203

where all the entries in the right side matrix are 2 • 2 matr ix operators acting all on the

same space H/ ' a ( ]R+) = ~rr = <2(1R+ . The symbols are

r O] r = 0 I

= A ~ A + = zz2 1 - - iZl t-1 1

where z2 = 0 in the decomposing case. Further we have

(4.104)

(4.105)

E1

W ~ ( W , , + I) - W,I.~o I - W.~oW~._o 0 - I 0 I 0 Wr_= - W ~ 1 - I 0 0 0

I 0 0 0 (W~I + I) W~_oC~, - W~_o. W ~ + I I I

W~_o~, I 0 0 0 0 0 I

0 0

- I I

(4.106)

(4.107)

with I = I[K~,2(1%)]2 and

91 = r = [ -~-1 /2( 1 - i z l t -1) - i z 2 t - l r -1 ] (4.108) r 1 - iz l t -1 [ ]

and z2 = 0 in the decomposing case. By construction [13], E~ and F1 are easily inverted. As a mat ter of fact, one has

W T - - a

I + W.~o W._o W T - - a

0

I

T_ a tTW

0

0 W.o (w., + I)

I 0

0 0 0 0 I 0 I - W . ~ - I - I 0 0 I

0 0 W,~ W.o

0 0 0 I

(4.109)

(4.11o)

In (4.103) the 4 • 4 block

w % = [ w~-~162 w~o~ (4.111)

can be inverted by the help of (4.91) and T(Ao) by inversion of (4.103), i.e.

( [ 1 ] ) T(Ao) -1 = R2• Y l I W,-~W;a+ wvW'*̂ _ 0 E11

0 I[~*.2(a+)]' (4.112)

204 Castro, Speck

using restriction R2x2 on the first 2 x 2 block. Therefore the inverse of (4.99) is

T - I (P§ + P_I[ )A ') r

= o H~,~(~)]~ +

-r' p -1 - P+AoP-) = A+ (P+AoP+ +

: - (4.113)

where E0 denotes the extension by zero on the complement [H~,2(11~_)] 2 followed by linear

extension onto [H ~,2 (II~)] 2. The calculation of I D-1 is completed by combining these formulas with (4.95) and (4.96).

ACKNOWLEDGEMENT: We like to express our gratitude to our guests Albrecht BSttcher, Yuri Karlovich, Ilya Spitkovsky, and colleagues Amelia Bastos and Antdnio Ferreira dos Santos for the fruitful discussions and helpful comments during the preparation of this manuscript.

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L. P. Castro Departamento de Matems Universidade de Aveiro Campus Universits de Santiago 3810 Aveiro Portugal

F.-O. Speck Departamento de Matems Instituto Superior T~cnico Avenida Rovisco Pals 1096 Lisboa Codex Portugal

AMS Classification Numbers: 47B35, 45E10, 47A20, 47A05, 46E35, 47A68, 78A45.

Submitted: July 17, 1999