Integral Equations and Operator Theory Vol. 13 (1990)
0378620X/90/0506713051.50+0.20/0 (c) 1990 Birkh~user Verlag, Basel
GENERALIZED
FACTORIZATION
OF NONRATIONAL
2 • 2 MATRIX
FOR A CLASS FUNCTIONS
A. B. Lebre" and A. F. dos Santos*
In this paper the g.eneralized factorization for a class of 2 x 2 piecewise continuous matrix functions on R is studied. Using a space transformation the problem is reduced to the generalized factorization of a scalar piecewise continuous function on a contour in the complex plane. Both canonical and noncanonical generalized factorization of the original matrix function are studied.
1
INTRODUCTION
In this paper we shall be concerned with the problem of calculating the generalized factorization (if it exists) of matrix functions of the form G =
[1
pla
Apa] 1 '
(1.1)
where A 6 C, p is a complexvalued function such that p~ is a rational function of the form:
P2(W) = (w~)w  k ~
,
k, kl E 9 \ R , n natural.
(1.2)
and a is an arbitrary piecewise continuous function on ~(= R U {oo}. In the context of WienerHopf factorization matrix functions of the form (1.1) with p2 rational are usually known as Daniele or Khrapkov matrices ([2],[7],[9]) and are quite common in problems from the areas of diffraction theory, acoustics and elastodynamics.
In recent years a great deal of effort has been made concerning the correct
formulation of some of these problems in a Sobolev space setting that, after reduction to L2 (R), lead to a 2 x 2 system of WienerHopf equations with presymbol of the form (1.1) (see, for instance, [4], [10] and [12]). We are then left with the problem of finding 9Sponsored by J.N.I.C.T. (Portugal) under grant no. 87422/MATM
672
Lebre and dos Santos
a generalized factorization of G on L2 (R) (see section 2 for the precise meaning of this concept). Because there is no general procedure to achieve a generalized factorization on L2 (R) of a matrix function of the form (1.1), these problems are usually treated by considering other kinds of factorization concepts, that nevertheless give less information for the problem in hand.
In [2] Daniele proposed a method to obtain what we call a
functiontheoretic factorization of G, i.e., the representation of G as G = GI G , where Gt, G , are matrix functions holomorphically extendable into the open lower/upper half plane II :F = {z E r : I m z ~ 0} and continuous on II:F. But in general (for p2 rational) the factors Gl,, have a non algebraic behaviour at infinity.
This is noted by Daniele
who in [3] proposed a change in his procedure to avoid this undesirable behaviour: he proved the existence of a rational matrix function R such that G = G t R G , where Gl,, satisfies the holomorphy properties cited above and have an algebraic behaviour at infinity. Although in some problems (i.e. for particular choices of p and a) it is possible to obtain a generalized factorization from a functiontheoretic one, this appears not to be feasible in the general case. For the case n = 1 in (1.2), taking a in an adequate class of functions it is possible to prove that Daniele's method does, indeed, yield a generalized factorization. This has been done by F. S. Teixeira in a recent paper [13]. For the case where p is itself a rational function, in a recent paper [8] one of the authors, using a method that bears no relation to the Daniele's approach, has achieved some progress in solving the problem of obtaining necessary and sufficient conditions for the existence of a canonical generalized factorization and , provided these conditions are fulfilled, derived explicit formulas for the factors. Also the key point in that work is the equivalence stated between the vector WienerHopf operator and a scalar operator, that apart from a finite rank operator, is the product of two very special scalar WienerHopf operators. (See also [9] for a different approach and where some references of applications to elastodynamical problems are given). In this paper a space transformation is used to deal with the case where p is not rational but satisfies (1.2). Through the space transformation an operator closely related to the WienerHopf operator with symbol (1.1) is transformed into another operator that can be expressed as the product of two scalar WienerHopf operators, following an approach similar to that used in [8]. Special attention is given to the case n = 1 and k = kl. The paper is organized as follows. In section 2 we recall some basic concepts and results from the theory of WienerHopf operators. For the case n = 1, we study in section 3 the connection between the basic singular integral operators acting on the L2
Lebre and dos Santos
673
spaces of functions defined on the real line and on a closed path (F) in the complex plane, obtained from the real line by a change of variable. Also in this section we introduce a scalar operator associated with the given vector WienerHopf operator, which facilitates the study in section 4 of the Fredholm property, indices and generalized invertibility of this operator.
It is also shown that the Fredholm characteristics of the WienerHopf
with presymbol G acting on a subspace of L2 (R) can be derived from the Fredholm characteristics of a scalar WienerHopf operator acting on a subspaee of L2 (r), whose presymbol g is easily constructed from the original one. For the case n = 1 and k = kl we derive in section 5 explicit formulas for the factors of a generalized factorization on L2 (R) of a matrix function of the form (1.1), whenever such factorization exists. (This includes both canonical and noncanonical factorization). The next section is devoted to a very important example from diffraction theory. In section 7 the following situation is considered: if the matrix function G with p of the form (1.2) belongs to the 2 x 2 matrix Wiener algebra on the real line, then the scalar function g defined on F, introduced in section 3 and used in section 5 to obtain the generalized factorization of G, belongs to a decomposing algebra of continuous functions on F, so that the factorization concept to be used, in this case, is that of factorization in an algebra. Finally, in section 8 we treat the case n 7~ 1.
2
PRELIMINARIES
Let us begin by introducing some notation. If X is a vector space then X~, n integer, denotes the set of all n  d i m e n s i o n a l vectors with components from X and X~x, denotes the set of all n x n matrices with elements from X. then a norm in Xn is defined by" IIX][x, = ~i~1 Ilxillx X is a Banach algebra then X~•
If X is a normed space
,x = (xl,...,x,~)
equipped with the norm
Ilxll
= nmax
C Xn, and if tlx,jilx
, x
=
{x,j}~j= 1 , is also a Banach algebra. Given a projection operator on X then a projection operator on X~ is defined eomponentwise, and we will use the same symbol to denote these two projections.
Let S~ : L2 (N) ~ L2 (it) denote the singular integral operator
defined by 8~r
1 ]~ a:o 1= 7ri
~(wo)dwo,
(2.1)
where the integral is understood as Cauchy principal value, which is known to be a bounded operator. Define the bounded projection operators on L2 (lt)
p•
= l ( I + S~)
(2.2)
and set / ~ (1~) = I m P • (this notation reflects the fact that the elements o f / ~ ( a ) are
674
Lebre and dos Santos
Fourier transforms of functions with supports on ~+).
Let ~ be the one point com
pactification of the real line in r U {oo}. By PC(~) we denote the algebra of piecewise leftcontinuous functions on ~. Let h E PC([~) and define the function h on ~ x [0, 1] by iz(a0,#) = h ( ~ ) + ~
[h6o+)  h(r
,
( ~ , ~ ) e ~ • [0,1].
We say that h is nonsingular (on Lz (~)) if the function 1~is never H e [PC(R)]~• and define the function Y on ~ x [0, 1] by .H(w,#)
= H(w) + #
[H(w+)  H(w)]
,
zero
on ~ • [0, 1]. Let
(w,#) C~[ x [0,11.
Then .~ is said to be nonsingular if the function h given by /t(w, #) =det H(w, #) , (w, #) E ~ x [0, 1] is nonsingular. It is well known that every nonsingular matrix function H e [PC(~)],•
admits a generalized factorization on L2 (R) (or relative to L2 (a)), i.e.
H can be represented in the form H = H_ D H+,
(2.3)
where (i)
D ~ diag {(r_Y;1) u' , . . . . (T_r~_l) un } r + ( w ) = c a  z+ , z+ C H + = { z E r : I m z
X
0}, u~ > ... _> u, integers, called the
partial indices of H, and ,.+1 ~rr:kl + 9 I t ,+ ( a ) ] . . . . . r1 . H•
9 [L + ( a ) ] . •
(ii) the operator H+ 1 P+ H 1
(2.4)
is bounded on [L2 (IR)]~ , or admits a bounded extension to [L2 (~)]~. If the partial indices are all zero the factorization is said to be canonical. For the sake of simplicity in (2.4) and in the rest of the paper we denote by the same symbol the operator of multiplication by a function and the function itself. Let C be a smooth contour in the complex plane. By this we mean that C is any counterclockwise oriented closed path which is sufficiently regular for the singular integral operator ,go : L2 (C) + L2 (C) defined by ScO(~)
1 [
1
= ~i Jc r  u
O(T) dr
(2.5)
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675
to represent a bounded linear operator (see, for instance, [11 or [11]). Let Pc, Qc be the pair of bounded complementary projection operators defined by Pc='~(I+Sc)
,
Qc=
(ISo)
(2.6)
and decompose L2 (C) = L + (C)@ L~(C) with L + (C) = Im Pc, L~(C) = ImQc. n~ (C) o
denotes the subspace of L2(C) of all functions of the form ~ = ~20 + c, where ~v0 EL~(C) and c is a constant. Let C be a smooth contour such that if r C C then  v E C. For this class of contours we define the reflection operator ,.7 : L2 (C) ~ L2 (C) by J~(T) = ~(r)
(2.7)
and denote by Pe, Po the pair of complementary projection operators Pe:l(I+J)
,
po=l(I,7)
(2.8)
associated with the decomposition L2 (C) = L2~ (C)  L2o (C) with L2~ (C) = I m P c,
L~o (C) = I m Po. When dealing with a vector WienerHopf operator it is sometimes more convenient to consider instead, a scalar operator closely related to it. This is done in the next leman, which is a different version of a similar l e m a n given in [8]. LEMMA 2.1 Let C be a smooth contour, G E [PC(C)]~•
of the form
and consider the WienerHopf operator 1Ca on [L2+(C)]~ with presymbol G,i.~., ~ c = Pc allmPc,
aZong
here Pc the projection operator of o, to let $ : L+(C) ~ L+(C) be the operator defined by s
= zPc  ec a Pc
F rther
,
where Ipc denote~ the identity operator on Im Pc. Then ] ( ~ G = A [ S0
I+0 ]
B
(2.9)
where A,B : [L+(C)]2 ~ [L+(C)]~ are the invertibte operators which together with their inverses are given by
0
IPc
TecbIpc IPc
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Lebre and dos Santos
P R O O F : By inspection.
Relation (2.9) means that )Ca is equivalent to ,9 9 I+. Thus, in particular we have (i) dimKerK;a = d i m K e r S
, codimlmKJa = codimlmS.
(ii) /Ca is a (semi) Fredholm operator if and only if S is a (semi) Fredholm operator. (iii) If S t is a generalized inverse (respectively left inverse, right inverse or inverse) of S then a generalized inverse (left inverse, right inverse or inverse) of K~a is given by )~ta~ B1 [ St0
I+01 A'
We shall refer to ,9 as the operator associated with ]Ca. REMARK : Lemma 2.1 remains true for functions G E [PC(~)]:•
We have
only to replace the projection operator Pc by the orthogonal projection P+ of L2 (~) onto L + (•). In this ease the WienerHopf operator ~ a is given by ]Ca = P+ GILr (~).
3
SPACE
TRANSFORMATION
AND
FIRST
RESULTS
As mentioned before, we shall use a change of variable in order to study the WienerHopf operator with presymbol of the form (1.1) for
P(~) = \ ~ + k~]
(3.1)
where, for definiteness, we choose k E ii + (the open upper half plane) and the branch of the square root function is such that p(w) E 1I+ for real w. Consider the transformation r = p(~),
(3.2)
that is a bijeetive mapping of the Pdemann sphere onto the union of the open upper half plane with the positive semiaxis Re(v) _> 0, and let
=6(~)=
k + kl v 2 1r2
(3_3)
Lebre and dos Santos
677
denote the inverse transformation. Let ro denote the image of the extended real line under the transformation (3.2) and define the operator C: L~ (R) ~ L2 (F0) by 1 C~(r)  1  r ~ ~(8(r)).
(3.4)
Let us show that C is a well defined and bounded operator. For ~, E L~ (~) we have
IIC~ll~=(~o~ = ]~ o I~(o(T))l = I1 
1  ~=1 = Id~I
21/,~ i~(,~)l=lp_,(~,)l
d~
J~l~(~)l~d~ = M t
_< M
II~IIL~<~>,
where M = 89sup,~e R Ipi(~)l. Moreover, C is also an invertible operator with inverse given by C_l,i>(~ ) _ k ++ kl k~ '~(P(")) (3.5) Let F denote the contour obtained by adding to Fo its symmetric with respect to the origin, so that a point r E F if r E Fo or  r E Fo. Note that F is the boundary of a simple or double connected region according to kl E H or kl E H +, respectively. Now let us consider the "even extension" operator e : L2 (Fo) + L2e (F) given by
e
~(r)
f~(r) 9 (r)
/
, ~er0 , r~r\ro
(3.6)
and define C~: L2 (~) + L2~ (P) by
C~ = eC,
(3.7)
where C is the operator defined by (3.4). The next result will be of great importance in all that follows. LEMMA 3.1 Let SR and Sr be the singular integral operators defined by (2.1)
and (2.5) with C replaced by F, respectively. Then (i)
(ii)
~
= C:'SrCe.
L~ (F) = P~L~ (F) = C~L~ (UR).
(3.S) (3.9)
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Lebre and dos Santos
PROOF : Note first that the subspace L2~ (F) is invariant under St. Then making use of the change of variable (3.2) we successively obtain:
&~(~) = ~ s
~(~o)d~o 0.) 0

02
k+kl
2 fro ~
p~(~) c~(~) d~
k+kl
i fr
1

~ + k7 ~i

~+k,~i
r
~p(~)G~(~)&
k+k~ (&G~) wbkl
(p(~)).
Using (3.5)
Now for any ~0i E / ~ (~) we have Ce ~
= • Ce Sn ~
= + Sv G ~ •
and so ff~+ = Ce ~0+ E L~ (F). Reciprocally, for any ~• E L~, (F)
C~ 1 ~+ = +C~l S r r • = •
Icb •
Thus ~ :  C[ 1 ff~• C L~ (~), and this completes the proof.** Let a be an element of PC(~) and consider the matrix function depending on the complex parameter A of the form G~ =
[lp la
Apa11 '
(3.10)
where p is the function defined by (3.1). Let 8~ : L2+(~) ~ L + ( ~ ) be the operator associated with/Ca~ via lemma 2.1, i.e.,
,~ = I+  AP+ paP+ p'aI+,
(3.11)
where I+ is the identity operator on L+ (~). We wish to relate the operator ,~\ with an operator on L + (F). Let us begin with the operator P+paI+. Using lemma 3.1 we obtain
P+ paI+ = 89(I + Sa) p a I + =~G1 l ( I + 8 r )
Gpai+
= C[1Pr Cr p a I+.
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679
Let ~ denote the function that coincides with r on P and let ao E PC(F) be the function given by ao(r) = / a(O(T)) , r E r0 (3.12) , T EF\Fo t  a(O(v)) (the subscript o stands for odd). Then
P+ paI+ = C[ 1 P r ~ a o C +, where C+ = CelL+(R)' In a similar way we get
P+paa I+ = C11 P r ao C+. Substituting these results in (3.11) we have, for I + denoting the identity operator on L~+ (C), 8~ = C[ 1 (I +  ~ Prao Prao I +) C+.
(3.13)
The operator ~ : L + (F) * L + (F) defined by ~ = I +  APrao Prao I + can be factorized as
= Pr(1 + v ~ ao) Pr(1  v/A aO)lL2+(r),
(3.14)
where the principal branch of the square root is chosen. Using this notation we have S~ = C[ 1 ~ C~+.
(3.15)
Let g~ E PC(F) be the function defined by = 1+
(3.16)
and denote by ~ and T/~ the WienerHopf operators on L + (F) with presymbols g~ and Jg:~, respectively, i.e., K~ = P r g~ IL2 + (r)
,
?/~ = Pr riga In2+(r)"
(3.17)
Then T~  ~
~
and ~,
~/~.
(3.18)
At this point it is convenient to emphasize some properties of the operators namely
(i) s
+,
(i{) ~=Y~Y,
({ii) ~ = ~
From these we deduce the following invariance property of ~r~ : ~(L+~ (F)) C L+~ (F)
,
T~ (L+o (F)) C L+o (F).
(3.19)
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Lebre and dos Santos
To prove this note that j:r~ = J~:~t~
= ~t~xJ
= ~:~J
= ~J.
(3.20)
But for Or E L+e (F),r E L+o(F) we have f l ~ = Or and flOo =  ~ o and from this property (3.20) follows. This property means that the operator T~ is completely reduced by the pair of subspaces (L+~ (F), L+o (F)) or, which is the same, that T~ is the direct sum of the operators Tx[Lr 4
and [Y~[Lr
FREDHOLM
'
PROPERTY, INDICES AND INVERTIBILITY
The objective of this section is to study the WienerHopf operator K:ax on [L + (R)]2 in regard to the determination of its Fi'edholm characteristics as well as conditions for invertibility and calculation of the inverse, if it exists. T H E O R E M 4.1 Let Y~G~ be the WienerHopf operator on [L+ (R)]2 with presymbol G~ defined by (3.10), ~
the WienerHopf operator on L + (F) with presymbol gx
defined by (3.16) and 7~ : n + (F) ~ L + (F) the operator defined by (3.14).
Then the
following propositions are equivalent: (i)
Ea~ is a Fredholm operator on [L + (n)]2.
(ii)
T~ is a Fredholm operator on L + (F).
(iii)
lCa is a Fredholm operator on L + (F).
Suppose that gx is nonsingular on L2 (F) and let u = indr g~ =  i n d EA. Then dim Ker Ea~ = max {0,  v }
,
codim ImtCax = max {0, v}
so that ind ]Ca~ =  v .
P R O O F : Let us prove first that (i) ~ (iii). Given GA E [PC(~)]2•
of the
form (3.10) we assign to it the matrix function G~ defined by G~(w,#) = ax(w) + p [G~(w+)  G~(w)]
,
(w,tt) C g •
(note that we are assuming that Gx is left continuous). It is well known from the general theory of WienerHopf operators (see, for instance, [1]) that /CG~ : [L + (R)]2 * [L + (~)]~ is a Fredholm operator if and only if GA is nonsingular on L2 (R), i.e., detG~(w,#)~0
,
(w,#)e~
•
(4.1)
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681
On the other hand, ~A is a laredholm operator if and only if gA E PC(F) is nonsingular on L2 (F) that is the symbol OA(~,#) = gA(~) + ~ [gA(~+)  gA(~)]
,
( , , ~ ) e r x [o, 1]
is such that ~x(r,#)#0
,
(r, #) E F x [0,1].
(4.2)
Thus we have to prove that (4.1) is true if and only if (4.2) is true. A straightforward computation gives
1A det GA(w, #) =
[a(a;)+#[a(w+)a(w)]]
(since we are assuming that a is left continuous: lim . . . . ga(r)
a(w)).
~
1  a [  a ( o ~ ) + ~ [a(o~+) + a(o~)l] =
,
(co, #) 6 ~ x [0,1]
,
~ e [0,1]
a(oc) = lim~,~+ooa(w), a ( o z + ) =
For the computation of 9=A we note that, by construction, we have
= 2  9a(r),
so that condition (4.2) is equivalent to 0h(r,#)r
,
(r,#)er0x
Recalling that F0 is the image under the transformation
=0~(p(w), #) =
[0,1].
(3.2) offi
we obtain
1~[a(w)+#[a(a~+)a(a;)]]
,
(~o,#)61%x [0,1]
1  V ~ [  a ( o o ) + # ( a ( o o + ) + a(oc)]]
,
p 6 [0, 1]
and it is now clear that (4.1) holds true if and only if (4.2) holds true. Now we prove that (ii) *~, (iii). If K~ is a Fredholm operator then also 7ta is a Fredholm operator and so is Tx, since T~ is the product of two Fredholm operators (Cf. (3.17) and (3.18)). Reciprocally, if Ta is a Fredholm operator then, in particular, T~ is a semiFredholm operator.
Now as T~ = Kx ~ a = 7/~ EA it follows that both /CA and
~ x are semiFredholm operators. But for a WienerHopf operator the properties of being Fredholm or semiFredholm are equivalent. Thus 7A cannot be a Fredholm operator if EA is not. Let us now prove the results concerning the dimensions of the kernel and cokernel of K~G~ if gx is nonsingular. Making use of lemma 2.1 and of formula (3.15) we have
dim Ker K~G~ = dim Ker Sa = dim Ker (T~ [g~ (rl) codim Im/CG, = codim
Ira&
= codim Im ( ~ IL~o(rl),
so we proceed to calculate the Fredholm characteristics of 7"a[L+ (r). It is well known that dim Ker ]CA = max {0,  u }
,
codim ImK~ = max {0, u}.
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Lebre and dos Santos
On the other hand, using the invaria~lce property of TA, we have Ker 7"/A = ,7 Ker/CA
,
Im 7/~ = ,JIm/CA.
Consider first the case v < 0. Then Im ~ = L + (F) and dim Ker T~ =  2 v , since indTA =  2 v . Let (I) E Ker TA; then T~(I) = 0 and T~,JO = 0 from which it follows that TA(I + ,J')(D = 0
,
T~(I  ff)(D = 0.
(4.3)
Now ( I + f f ) ~ 6 L+e (P) , (I  f f ) ~ e L+o (F); hence Ker T~ = Ker (2rAIL+~(r))ff~ Ker (2r~lL2+o(r)) . Relations (4.3) also imply the existence of a bijection between Ker (TA In2*~(r)) and Ker (T~ 1%+o(r)) and consequently d i m K e r ('/AIL+o(r))= dimKer (~AlL+o(r)) =   v . For the case v _ 0 we begin by noting that Ker T~ = (0) and codlin Im ~ = 2v since indT~ = 2v. Let k0 e I m p ; then there exists (I) 9 L + (F) such that ~
= o2, from
which it follows that ~ff(I) = flk~. Thus 9 Im ~ ~ , J ~ E I m ~ .
(4.4)
Hence there exist subspaces Me and Mo such that L + ( F ) = (Im (T~[L2+,r) ) * Mr) * (Im (T~[/+o,r)) * M o ) . But (4.4)implies that there exists a bijection between Im (T,~]L+ (r)) and Im ('TA]L+o(r)) which, in view of the above decomposition, implies the existence of a bijection between M~ and Mo. Thus dim M~  dim Mo = v i.e. codlin ha
thus completing the proof of the theorem. 9
Lebre and dos Santos
683
We conclude this section with a remark on the nature of the operator/Cv. R E M A R K : T h e equivalence of propositions (i) and (iii) signifies that the vector WienerHopf operator K:G~ is always left invertible or right invertible as for WienerHopf operators with a scalarvalued symbol. Indeed its invertibility is determined by the scalar function g~, This is a rather unexpected result which is further exploited in the following section. 5
EXPLICIT FACTORIZATION
In this section we are going to consider the problem of obtaining the generalized factorization on L2 (R) ( whenever it exists) of the matrix function G=
[
Xpa] 1 '
1
pla
(5.1)
where A E ~, a 6 PC(K) and p is the function defined by (3.1) with k = kx 9 Here we have omitted the subscript A in G for the sake of simplicity. Making use of the transformation (3.2) let us consider the matrix function 6 G [PC(F)]2• defined by
6(T) =
{
a(0(T))
,
T9
G(O(T)
,
~ 9 r \
r0,
(5.2)
i.e., in the notation of the preceding section 6(T) =
[ lr_lao(r)
X~'ao(7) 11
'
(5.3)
Let us begin by considering canonical factorizations. Suppose that 6 admits a canonical generalized factorization on L2 (F), that is 6 can be represented as 6=6
6+,
where (i)
6+• 9 [L+ (F)]:•
,
6~ 1 9 [L~ (F)]2•
(ii) the operator 6+1Pr6_1 is bounded on [L2 (F)]2• or admits a bounded extension to [n2 (F)12• As by construction 6 is an even function on F it follows that 6+ satisfy the same property, i.e., G+ 9 [L~ (F)]2• From (3.9)it follows that C~1 6~ 9 [L~ (R)12x2.
(5.4)
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Lebre and dos Santos
So if we define
a~(~)=
w+k
2k c:'a~(~)=a~(p(~))
(5.5)
we have a = G_ G+.
(5.6)
Let us show that this factorization is a canonical generalized factorization on L2 (!~). From (5.4) we conclude that r~V~eL~(a)
,
for
1
r•
and to prove that G+ 1 P+GTJ defines a bounded linear operator on L2 (R) it is sufficient to note that
a;' P+a:' = c : 1 (~'PF~:') ct It is now clear that the above reasoning can be reversed, i.e., from a canonical generalized factorization of G we can obtain a canonical generalized factorization of G. The advantadge of considering the factorization problem for G instead of G is that effective factorization methods are available for this class of matrix functions (see
[8D. Let us write
0,T= I100[11 ]
0]
(5.7)
with H=
[ lv/~ao v~ao]l "
(5.8)
Concerning the existence and determination of a canonical generalized factorization of the matrix functions of this form we have the following result : THEOREM 5.1 Let C be a smooth contour and B e [PC(C)]2• B=
[ 1b ] b 1
of the form
"
(5.9)
Then B admits a canonical generalized ]actorization on L2(C) if and only if the functions 1 + b, 1  b admit canonical generalized [actorizations on L2(C) . Moreover, if
1 +b=t_
t+
,
1b=~_
~+
(5.10)
Lebre and dos Santos
685
are canonical generalized factorizations on L2(C), then a canonical generalized [actorization of B is given by B_ = ~
' t_s_
=2
t_+s_
. (5.11) t+s+
t++s+
P R O O F : The proof of the first part can be done through a reasoning similar to the one used to prove theorem 4.1 , having instead of the WienerHopf operator with presymbol B, the scalar operator associated with it , which, in this case, is the product of two WienerHopf operators with presymbols 1 + b and 1  b ( see also [8], theorem 2.3 ). The proof of the second part can be done as follows : obviously B+ e [L~ (C)]2x 2 and a straightforward computation shows that B = B_ B+.
Finally to prove that
B+IPcB_ 1 defines a bounded linear operator on L2(C) or an operator with a bounded extension to L2(C) we consider the matrix representation of this operator
B+IPcBz 1 = ~ lg Y
lg + Y
with
U~=t+ 1Pct:l~
,
Y ~ = s + 1Pc's: 1
that are bounded operators on L2(C), or admit bounded extensions to L2(C), since (5.10) are canonical generalized factorizations ... The matrix function H belongs to the class considered in the above theorem and has the additional property that the function that appears in the second diagonal is an odd function on F, which implies that S+(T) = t+(T) and consequently t•177
=2Pet+
,
t•177
=2Pot:~.
Making use of this relations we have: COROLLARY 5.2 Let a~ E PC(F) be such that a o (  r ) =  a o ( 7 ) , r E F.
Then H admits a canonical generalized factorization on L2 (F) if and only if the function g l+v~a~
(5.12)
admits a canonical generalized factorization on L2 (F). Moreover, if g = g  g+
(5.13)
686
Lebre and dos Santos
ia a canonical generalized ]actorization, then a canonical generalized factorizatwn of H H_ H+ is gwen by
H
= [P{g_F;g_ P~gP{'g], H+  [P,g+pog+ P~g+P('g+] .
(5.14)
REMARI,: : It should be noted that the first part of this result had already
been proved in theorem 4.1, since a WienerHopf operator is invertible if and only if its symbol admits a canonical generalized factorization. In this context it should also be noted that the operator 72, appearing in section 3, is the scalar operator associated with the \VienerHopf operator with presymbol H. We are now able to state the main restilt concerning canonical factorization of the inatrix function G : THEOREM 5.3 Let k C r 6 PC(~) and p the function defined by (3.1) and let G and ~ be the matrizvalued function3 defined by (5.1) and (5.3), re.~pectivety. Then G ( re,spectively G) admit~ a canonical generalized factorization on L2 (~ ) ( L2 (F)) if and only if the f.,n~:tzon
g = I + v/A ;q,
(5.15~
adrait,s a canonical generalized factorization on L2 (F). Moreover, if
g = g_ g+
(5.16)
i,s a canonical generalized factorization on L2 (F) then a canonical generalized factorization ~ 6+ of g on L2(F) i,~ given by
~P{(r 'g)
Pj]
~.~P~( t  ' g + )
Pc,q+
l
(5.17) and a canonical generalized .faetorization G_ G+ of G on L2 (~,) is given by G_(~.,) =
g_(p(w))
,
0+(,.9
= g+(p(~o)).
(5.18)
PROOF : The first par t is a consequence of (5.7) , theorem 5.1 and corollary 5.2. Suppose now that g admits a canonical generalized facto~ization on L2 (F), say g = g_ g+. By corollary 5.2 tim matrix function H, defined by {5.8), admits a canonical generalized
Lebre and dos Santos
687
factorization on L2 (I~), say H = H_ H+, with H• given by (5.14). We now show that the matrix functions
o
o v%
Pog•
(5.19) define a canonical generalized factorization of G. First it is obvious that ~_ ~+ = ~. On the other hand, for the functions in the second diagonal of (5.19) it is not difficult to see that P~(Tg+) , P~(r'g• So ~• C [L~ (F)]2•
e L ~ (r).
It remains to prove that ~+1Pr~21 defines a bounded operator oi1
L2 (F) or an operator with a bounded extension to L2 (r). As before, if we use the matrix representation of this operator, after some computations we arrive at 9 + 1 P r 9 _1 = 1
[U+F+J~ UI)+&] UV+Ja
/2+ V+J4
where ,7i, i = 1 , . . . , 4 are finite rank operators on L2 (I') and/g, 12 are given by
Ll~=g+'Prg2'~
,
12~=(flg$1)pr(flg21)~.
U and 12 are bounded linear operators on L2 ( r ) or admit bounded extensions to L2 (F), since 9 = 9 9+ is a canonical generalized factorization. We have thus obtained a canonical generalized factorization of 9 and, as we have shown at the begining of this section, if we define the factors G• of G by (5.18) we obtain a canonical generalized factorization on L2 (IR) of G . This completes the proof of the theorem.m Our next objective is to describe all possible factorizations of the matrix function G. As regards the invertibility properties of the WienerHopf operator /Ca we have shown in theorem 4.1 that, supposing 9 nonsingular on L2 (F), there are only three possibilities: /Ca is left invertible, right invertible or invertible. In terms of factorization on L2 (PQ this means that the partial indices of G, say ul and v2, are always of the same sign and, moreover ul + u2 = u = i n d r g .
688
Lebre and dos Santos
We show next that the partial indices are as close as possible to each other, i.e., //~ ~,~ = ~ ~ if 1//t is even, and 1/1
//2 q 1 = ~ 2 , if 1~'] is odd. Let us begin bv~ obtaining a
~
factorization of H on L2 (F) from a given factorization of g. LEMMA 5.4 Suppose g i~ nonsingular on L2 (F) and let g =g_ r"g+
be a generalized factorization of g. Al~o let Hi be the matrix functions defined by (5.1/,) and define
D(r) = diag { r ~, r " } .
(5.20)
Then a generalized factorization of H on L2 (F) is given by H = H_ D H+,
(5.21)
'0/here H+
if I//I is even
[ 0H 0+ 1 ]I
if t//]is~
H+
(5.22)
P R O O F : We have H_ D = r " H _ . Let us show that H_ i~I+= r" H.
A simple computation gives
H_
H+
=
[ P~(g_g+)
[ Po(gg+)
Po(gg+) ] K(gg+)
[
=
P,(T~F)
Po(rL~J) ]
Po(r~g)
Pr
j "
Taking into account that
P~(,
tog)
[
/
~~
, if
tvt
is even
Po(<~g) v/~ r  " a o , if ),[ is odd
we conclude that for
I//I
= { T, /~2 r~ao
, if F'I i~ even
. if ]//i is odd,
even
H_ fl+ = H _
H+
= r ~'
z Tz' H
1
ao
I 1 g/~a~
Lebre and dos Santos
689
and for ju I odd
1][01] v'~ao
= r~la H.
1 0
Thus for any value of the integer u we have H_ H+ = r  " H, as we had to prove.n We can now state the main result concerning noncanonical generalized factorization of G. T H E O R E M 5.5 Suppose g is nonsingular on L2 (P) and let
g = g _ r"g+. Also let ~+ be the matrix functions defined by (5.19) and define
DI=
diag{r",r"}
,
if lul is even
diag{'"+', r"'}
,
if lu[ is odd
(5.23)
Then a generalized factorization of ~ on L2 (P) is given by G = Q _ D, r
(5.24)
where
G+
,
if luIiseven
[ 0. G +01l]
'
if [ u l i s ~
(5.25) ~+=
and a generalized factorization on L2 (~ ) of G, G = G_ DG+, is given by G_(w) = G_(p(a.,))
,
G+(w) = ~+(p(w))
,
D(w) = D,(p(w)).
(5.26)
P R O O F : Let H = H_ D H+ be the factorization obtained in the last lemma. Thcn G = G _ DF~, where _G_ is given by (5.19) and F~, is also given by (5.19) with H+ replaced by I~+. Now for even lul we have F\ = G+, since I~+ = H+. So if we take Dl = D ageneralized factorization of G is obtained. If lul is odd, in general F, r .+ (F)]~• but we have 0 T 1
1 0
0
r'
Lebre and dos Santos
690
In this case a generalized factorization is obtained by g=g_ with
r
DI=D
0
0 ]
D, 0+
= diag {r"+l,r"l}.
T I
Noting that ( ~  k ~ f , (~,b'Tgl ~  k ) ~ ] j, diag [, [ \~,+~/
if Ivl i s e v e n
D(~,) = D,(p(w)) = diag [ \ ~ + L ,
' \77g]
j
,
if ]u I is odd
it is easy to conclude t h a t G = G DG+, with the factors defined by (5.26), is a generalized factorization on L2 (JR). 9 R E M A R K S : There are some facts about the factorization p r o b l e m for a matrix function G of the form (5.1) t h a t nmst be emphasized: 9 First of all the existence of a generalized factorization for G on L2 (~R) is reduced to the existence of a generalized factorization of the scalm' flmction 9, m an L~space of fimctions defined on a different contour, L2 (P). 9 W h e n e v e r a generalized factorization of G exists, the factors G~: of the factorization o b t a i n e d in theorem 5.5 alw~gvs conmmte with each ()ttwr. 9 The factorization of G on L~ (IR) is a b()un(h'd fa.('t(nizatiozL i.('., G:~ arc bounded on [q , if and only if the fa('torizati(m ()f g is a botutdcd one (s('(' how the factors G~ a.rc o b t a i n e d fiozn the factors of ~.t). 9 The results of this section can be used to l)row" tim f()llowing generalization ()f a theorem of F.S. T('ix('ira (so(: {13], theorem 3.6) T H E O R E M 5.6 Let G be a matrzz fltnction of the f o r m (5.1) unth A ~ ~, /)
d4i,.,;d
@ (.'~.1) ,~,,l ,, c
C(fi). 1
S,,.pp,,.,,: ~/,.,,.~,
~/~,,(x)(21,
 1) r o
.
o,v,d ttl.a,f the f,...(:t, zov.~ dl ,,.d (l~ d@)~,cd by ,/,(~:) a,r~
: 1 + ~ / L , ( ~ . , )
4(0:)
=: 1  v ~ , , ( ~ . . )
.
~.. ~ ~
.~u.ch that indwl} indt~d~ ~ 0 .
(5.28)
Lebre and dos Santos
691
Then G admit~ a generalized canonical factorization on Lz(R), G = G_G+, where the factors G+ are given by Daniele's formula~, i.e.
~p~
c+
with d~ = (dl•189
,
c:~ = cosht+
,
s~ = sinht:~
for dl = dl_d2+ and d2 = d2_d~+ denoting generalized canonical factorization of dx and d2 on L2([~) and t:~=
tiP+
log
,
/~(w)=(~k) 89189
, wen.
Sketch of the proof: First note that, for a E C(~), conditions (5.27) and (5.28) are sufficient to garantee that the function g, defined on F by (5.15), admits a canonical generalized factorization on L2(F). Thus, by theorem 5.3, G admits a canonical generalized factorization on L2(R). The rest of the proof consists of algebraic manipulations of the factors given in that theorem. 9
6
EXAMPLE
FROM
DIFFRACTION
THEORY
In this section we shall apply the theory of the preceding sections to an im portant example from diffraction theory recently considered by F.O. Speck [12]. This example corresponds to the matrix function (5.1) with a = I and k = kl, i.e.,
pi
1
'
where A E r and p is the function defined by (3.1) with k = kl. Using theorem 4.1, we know that a generalized factorization of G on L2 (R) exists if and only if the scalar symbol g = 1 + vfAa with a(r)= is nonsingular on L2 (F).
1 1
, ,
r E F0 Ter\r0
A further look at the proof of that theorem shows that a
generalized factorization exists if and only if A E 9 is such that 1 + V/A~(T) + p [~
 a(r)] ~ 0,2
,
(T,~) e ro • [0,1]
692
Lebre and dos Santos
or
r {1, ool. On the other hand, for these values of A, we have u = i n d r g = 0, so we conclude that G always admits a canonical generalized factorization for A r [1, oo[. Moreover, the factors of a factorization, given by the formulas in theorem 5.3, are determined once we have obtained a canonical factorization of g. To this end let
l+vq a =
log 1   v ~ '
where the principal branch of the logarithm is considered, and define g(~)
V+IV
= ~7~1z_
'
9+(r) = ~_
1/§
'
where z ~ _ and z~_ are the branches of z" with branch cuts on Ft and P~+, respectively. Then, noting that for the choice made for the branch of the logarithm we have, for A r [1, oo[, 1 
1 2
from which it follows that g+ e L~ (F). Now from g_(T)g+(r)=
1
,
IrnT>O
e2,~i,~= ~ 1  ~
,
ImT
we obtain:
= (l + v q ) g_g+. To conclude that this representation is a canonical generalized factorization of g on L2 (F) (the constant 1 + V ~ can be associated with any one of the factors g~ ) it is sufficient to recall that the projection operator P r is a bounded linear operator on the weighted space
L~(r, ~) ={r meas~able onr: ~r ~ L2 (F)} with weight T1
Rea
1
For the present example there are some additional properties of the factors g+. In particular g_(r) = g'(T)
,
g+(T) = e 2'~~ gT'(r),
Lebre and dos Santos
693
so that we m a y write
1 Pc g = ~ ( g  + g51) = cosh log g _ 1
. . . .
i~
1 ,~i~,. ,~i,~ eog+~e(e g +  e_,~i,~9+l )  x/1 l ~ /~ sAm n.t o. g. ( .e
is
P, 9+ = ~ e 
~i~" ~i~
1
Po g_ = ~(g_  g_') = sinh log 9 _
,
(e
g+ +
e_~i~g+l ) = x/1  ~
1 + v ~ e~176
,
g+)
,
g+).
Introduce now the functions
8 ~=Z+ "7+ ~
,
where/3+ are the branches of ( w + k ) { with branch cuts C+ = {z e r : z + k Then
g(P)=~p1]_
:~
= ix, x E R~:}.
=\~7
and e,~i,~ g + ( p ) =
+
2~ = '7+ 9
With this notation the substituition of the above results in formulas (5.18) gives for the factors of the canonical generalized factorization of G on L~ (R), for A r [1, c~[, G+ = (1  A)~ [
[ and # = 2a =
cosh(plog'y~)
x/~p sinh(#1og3,•
]
1 p1 sinh(plog~+)
cosh(#log3,+)
J
1 I o g 1_2~" 1+4~ ~7
It should be noted that the factors of the factorization of G just obtained are given by Daniele's formulas (see theorem 5.6). As mentioned above, the factorization problem for G was considered before by F.O. Speck, who obtained a distinct factorization of G on L2([~), say G = G _ G + (whenever it exists). It can be shown that a constant matrix K exists such that G_IG 1 = G+G+ ~ = K
In fact, K is given by
for some a E (I:. The existence of such K is, of course, a consequence of the uniqueness of the canonical factorization.
694
Lebre and dos Santos 7
FACTORIZATION IN THE WIENER ALGEBRA
In this section we are concerned with the following situation: let )IV denote the Wiener algebra on the real line and suppose that a E P C ( ~ ) is such that the matrix function G defined by (5.1) belongs to 14;2•
that is, pa C IN. Then, as is well known, a
generalized factorization on L2 (•) of G is actually a Byfactorization, i.e. the factors G:k of the representation (2.3) are such that G+~1 C 14;2+X2
,
G_~1 E IN2x2,
where W + = IN A C+([[), C + ( ~ ) denote the subalgebras of continuous functions on I~ possessing an analytic extension into the upper/lower half plane. However, from the theory expounded in the preceding sections, that factorization is related to the generalized factorization on L2 (F) of the scalar function g, defined by (5.13). We next show that there exists a decomposing algebra .M of continuous functions on F to which g belongs if G E BY2x2, so that the Byfactorization of G is related to the factorization of g in the algebra .M. Let A : IN ~ C(F0) be the operator defined by .Ac2(r) = c2(0(T))
(7.1)
and let .4e = e A, where e is the even extension operator defined by (3.6). Consider now the set M={~cC(r):~=~ae~,+f~ae~,l
,
~,,~,CIN}
(7.2)
and introduce a norm in M by
I1~11~ = ~(ll~llw + Ill, lily), where a = I I p ~ l l ~ > I. LEMMA 7.1
(1) M is a Banach algebra. (2) 3/1 is an Ralgebra. (3) Ad is a decomposing algebra.
P R O O F : (1) M is obviously a Banach space. Let us show that M is a Banach algebra. Let ~, ko be arbitrary elements in M and write
Lebre and dos Santos
695
with ~0,~o1,r162 E PV. T h e n
= Thus r
A,(~r
+ p%~r
+ §162
+ ~r
E A4, since the terms in brackets are elements of ~Y. Moreover
11O~ll
=
a(ll~r + p ~ , r
< ~(ll~llllr =
+ I1~,r + ~r
+ tl~,llllCxll + II~llllCll + tl~llllr
~(11~11 + I1~,11)(11r
+ t1r
= Itr
so t h a t .M is a Banach algebra. (2) It is easily seen t h a t AA contains the set T~(F) of all rational functions having no poles on F . Let us show t h a t T~(F) is dense in f14. Let r E .M and write r = A~o + ~ A ~
(7.3)
for some ~o, ~01 E }iV. Because ]/Y is an R  a l g e b r a we can choose r, rl rational function having no poles on ~ such t h a t
are as small as we want to. Defining
R : A~r + r we conclude t h a t the n o r m of r  R = A ~ ( ~  r) + e A ~ ( ~ ,  r,) can be m a d e a r b i t r a r i l y small. (3) To show t h a t A4 is a decomposing algebra it is enough to show t h a t the singular integral o p e r a t o r S r is b o u n d e d on A4. Let r E A4 and use its r e p r e s e n t a t i o n as in (7.3). T h e n
Sr'b
= Sr A ~ + Sr ~  A ~ = SrA~+~SrA~
= CeS~r~ + ~C~S~r~I
696
Lebre and dos Santos
where r(w) = ~~. 2k As is well known the operator defined by
,~
= r' /.q~r~
(7.4)
is a bounded operator on W. Then
IlSr~ll~ = II&~llw + II&~Hw < M(tl~llw + II~'lllw) = M~ II~IIM for some M, M1, as we wanted to prove.,, Define as usual the subalgebras M e of 34 by
344 = 34 n C• where C+(F) denotes the subalgebra of all continuous functions on I" possessing an analytic extension into the region interior/exterior to F. Because 34 is a decomposing algebra of continuous function on P we have M = 34+ ~ 34o in which 34o denotes the subalgebra of 3 4  consisting of those functions 9 that satisfy ~(oo) = 0. Moreover, the projection Pr = ~(I + 8r) coincides with the projection of 34 onto 34+ along M o. As is well known W is a decomposing algebra of continuous functions on ~R : W = W + ~Wo, where W o = {9~ E W  : c~(a) = 0, a E Hfixed}. Moreover, the projection operator of W onto W + along Wo is given b y / 5 + = 89 + ~ ) , for , ~ denoting the operator defined by (7.4). Using the connection between 8r and , ~ established in the proof of the preceding lemma we easily obtain the following characterization of the subalgebras 344 : 344 =
ctr):
9 =
+
Ao ,,
,
w4}.
Since 34 is a decomposing Ralgebra of continuous functions on P, every invertible element rn E M admits an 34factorization, that is, rn earl be represented as T rt ~
TTZ_ 7~ u
T~+
with ,n+~ E M + ,/52~1E 34 and L, = indrm (see [1]). Futhermore the factors m• of an Adfactorization can be explicitly evaluated by m+ = e x p P r l o g ( §
,
m_ = e x p ( I  P r ) l o g ( ~  " m ) .
Lebre and dos Santos
697
Returning to the question of how to determine the factorization of the matrix function G E W2xz, i.e., when a is such that pa E W (or what is equivalent pZa E W, since p2 E W) we have g= l+v~a~
e M.
In fact
g = 1 + v ' ~ ~'a~ = 1 + v / ' ~ A,(pZa). Thus if pla E W then g E M . Consequently if g is invertible in M , an A~factorization of it can be obtained. Noting that the subalgebras M + and A4 are closed under the operation of multiplication by § and ~.1, respectively, it is not difficult to see that all the results given in section 5 for the factorization of G and G are correct if we substitute generalized factorization for Mfactorization. 8
GENERALIZATION
In this section we shall generalize the results of the preceding sections to matrix functions of the form G =
[lp_,~a
where n is a natural number and as before A E r
~p'~a]l '
(8.1)
p is the function defined by (3.1) and
a is an arbitrary piecewise continuous function on ~. As mentioned in the introduction, the case where n is an even number was considered in [8], in which although a is assumed to belong to the Wiener algebra on the real line, all results can be generalized without difficulty for a E PC(R). Therefore in what follows we suppose that n is a natural odd number. Let us consider the operator S~ associated with the WienerHopf operator via lemma 2.1, i.e.
S~ = I+  %P+pn a P+ p'~ a 1+. We have
P+p'~ a I+ = C / 1 P r ~ " a~ C+, Now, making use of the decomposition 1
1
~(~ ~)
~"(~ ~)


k=l T k u n  k  1
'
T, It E F,
we get
P+ p" a I+ = C['r "n P r ao C+~  Cg ~ ~ ~k'~*ak(C+ ), k=l
698
Lebre and dos Santos
where ok, k = 1 , . . . , n, are bounded linear functionals on L2, (F) defined by ak(O,) = ~ri
~  ao qS, dr
It should be noted that for ~ E L2r (P) ak(a2~)=0
ifkisodd.
Using these results we arrive at _._& ~ 2
8~ = C~~ (I +  k Prao Pcao I +) C+  C;~Prao ~
r 2k~ a ~,~'+'j
k=l
or
s,, = c 2 ' ( %
 &)c{,
where "Tx is the operator defined by (3.14) and Jx : L+~ ( r ) ~ L~~(F) is the finite rank operator defined by ....._~1 n 2
~ 0 ~ = APrao ~
r 2k1 a2k(q~ ).
k=l
We are now in position to state a theorem which gives necessary and sufficient conditions for the matrix function G in (8.1) to admit a canonical generalized fac~orization. T H E O R E M 8.1 Let G E [PC(I~)]2•
be a matrix function of the form (8.1)
with n a natural odd number. Then necessary and su.t~cient conditions for G to admit a canonical generalized factorization on L~ (~) are : (i) The scalar function g~ E PC(F) defined by (3.16) admits a canonical generalized factorization on L2 (F), (ii) The finite rank operator
I~;' & is invertible. P R O O F : The proof is extremely simple if we note that proposition (i) is equivalent to any one of the following : (i')
T:~ is invertible ,
(i")
indTa = 0 ,
Lebre and dos Santos
699
which are consequences of the representation of ~r~ as the product of the WienerHopf operators with presymbols g~ and g/'g~.i The canonical generalized factorization of G (if it exists) can be obtained using a procedure similar to the one considered in [8], section 3.
REFERENCES
[1]
Clancey, K. and Gohberg, I. : Factorization of Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications, vol. 3, Birkhs Verlag, Basel, etc. ,1981.
[2]
Daniele, V. T. : On the factorization of WienerHopf matrices in problems solvable with Hurd's method, I.E.E.E. Trans. Antennas Propag., vol. AP26 (1978), 614616.
[3]
Daniele, V. T. : On the solution of two coupled WienerHopf equations, SIAM J. Appl. Math., vol. 44, No. 4 (1984), 667680.
[4]
dos Santos, A. F., Lebre, A. B. and Teixeira, F. S. : The diffraction problem for a half plane with different face impedances revisited, :l. Math. Anal. Appl., vol. 140, No. 2 (1989), 485509. Gohberg, I. and Krein, M. G. : Systems of integral equations on a half line with kernels depending on the difference of arguments, Uspehi Mat. Nauk. 13, No.2 (80), 372 (1958). English Translation: Amer. Math. Soc. Trans. (2) 14, 217287 (1960).
[6]
Hurd, R. A. : The explicit factorization of 2 x 2 WienerHopf matrices, Preprint Nr. 1040 Fachbereich Mathematik, Technische Hochschule Darmstadt, 1987.
[7]
Khrapkov, A. A. : Certain cases of the elastic equilibrium of an infinite wedge with a nonsymmetric notch at the vertex, subjected to the concentrated force, Prikl. Mat. Mekeh., 35 (1971), 625637.
Is]
Lebre, A. B. : Factorization in the Wiener algebra of a class of 2 x 2 matrix functions, Integral Equations and Operator Theory, 12 (1989), 408423.
[9]
Meister, E. and Speck, F.O. : WienerHopf factorization of certain nonrational matrix functions in mathematical physics, in: The Gohberg Aniversary Collection, Operator Theory: Advances and Applications, vol. 41, Birkh~user Verlag, Basel etc., 1989, 385394.
[10]
Meister, E. and Speck, F.O. : Modern WienerHopf methods in diffraction theory, in: Ordinary Diff. Eqs., vol.2 (ed B. Sleeman and R. Jarvis), Proe Conf. Dundee 1988, Longman, London 1989, 130171.
[11]
Mikhlin, S. G. and PrSssdorf, S. : Singular Integral Operator, SpringerVerlag, Berlin, 1986.
[12]
Speck, F.O. : Sommerfeld diffraction problems with first and second kind boundary conditions. SIAM J. Math. Anal., 20 (1989), 396407.
700
Lebre and dos Santos
[13]
Teixeira, F. S. : Generalized factorization for a class of symbols in [PC(~)]2• Appl. Anal., vol. 36 (1990), 95117.
Departamento de Matem~tica Instituto Superior T~cnlco Av. Rovisco Pals 1096 Lisboa Codex Portugal
Submitted: Revised:
or
August 15, 1989 March 23, 1990
Centro de An~lise e Processamento de Sinais Complexo I do I.N.I.C. Av. Rovisco Pals 1000 Lisboa Portugal