Integr. equ. oper. theory 36 (4) 409-432 0378-620X//040409-24 $1.50+0.20/0 9 Birkh~user Verlag, Basel, 2000

I IntegralEquations and OperatorTheory

W I E N E R - H O P F F A C T O R I Z A T I O N F O R A CLASS OF OSCILLATORY SYMBOLS M. C. Cs

A. F. dos Santos

Two classes of 2 x 2 matrix symbols involving oscillatory functions are considered, one of which consists of triangular matrices. An equivalence theorem is obtained, concerning the solution of Riemann-Hilbert problems associated with each of them. Conditions for existence of canonical generalized factorization are established, as well as boundedness conditions for the factors. Explicit formulas are derived for the factors, showing in particular that only one of the columns needs to be calculated. The results are applied to solving a corona problem.

1

INTRODUCTION

In this paper we study the Wiener-Hopf factorization for two classes of oscillatory 2 x 2 matrix-valued functions. Existence of canonical generalized factorization is examined and explicit formulas are derived for the factors. The motivation for the choice of these classes comes from the Riemann-Hilbert problems that appear in "inverse-scattering" [1] and from the theory of convolution operators on a finite interval. The apparent simplicity of the first of the above classes studied is deceptive since it turns out that, except when the functions involved are a combination of exponentials and rational functions, no general method exists for obtaining conditions of existence of canonical factorization, much less for calculating the factors. The analysis presented in the following sections is far from applying to any symbol of the general form that appears in one-dimension inverse-scattering [1], but the authors feel that the method proposed can be applied, with appropriate modifications, to more general classes of symbols. Some of the ideas used in the present paper have their roots in earlier papers by the authors, where the aim was to study the Wiener-Hopf factorization of other classes of symbols that are motivated by applications in Diffraction Theory ([6], [7]). In particular, it is shown that the calculation of the factors reduces to the determination of a single column. An interesting result of the paper is an equivalence theorem which relates the factorization problem for the two above-mentioned classes of symbols. This imediately shows that the factorization of symbols of the type that appears in inverse-scattering is a difficult problem bearing in mind the difficulties posed by the factorization of triangular symbols of the type that appear in connection with finite interval convolution operators. On the other hand, the work reported in this paper has convinced the authors

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C~nara, dos Santos

that Riemarm-Hilbert problems with appropriate coefficient functions can be successfully used to obtain solutions to certain corona problems. This is a perspective opposite to that taken in [3], [41 and [51 and may be adopted by the authors in a subsequent paper. 2

PRELIMINARIES

We begin by defining the spaces of functions which will be used later. Let F denote the straight line contour R + iA, A E R, oriented as the real line. We denote by Lp(F), 1 _< p < oc, the Banach space of all complex-valued Lebesgue measurable functions defined on F, for which If] p is integrable, with the norm "f'[P = ( f r If(t)I' dt) 1/p

(2.1)

The singular integral operator Sr : Lp(F) -~ Lp(F), 1 < p < oo, is defined as usual by 1

Srf(t) = in fr uf(u) - t du

t EF

(2.2)

where the integral is understood in the sense of Cauchy's principal value. Related to this operator we define two complementary projections P~ = ~(I + St) ,

(2.3)

where I is the identity operator in Lp(F). For F = R, in particular, we take P~: = P• We denote by L+(F) and Lp(F)) the images of P+ and P r , respectively. Defining L~(F) as the space of all essentially bounded functions f : F --~ C, we denote by L+(F) (resp. L~(F)) the subspace of those functions f E L ~ ( F ) which admit a bounded analytic extension to the half-plane C+ + iA = {z E r : I m z > A} (resp. r162 By 8~(L~(F)), a E R, we denote the space of functions f E L2(F) such that f(~) = ei~r177 with f• E L~(F). It is clear that, for a > 0,

P+f E E~(L~(F)), if f E C.(L~(F)) ,

E_.(L~+(r)), E.(L~(F)). Prf E

if f E

C_.(L+(r))

(2.4) .

(2.5)

We define analogously If A is an algebra, let ~(A) be the group of invertible elements in A. By a generalized factorization of G E 6(Loo(a)) 2• (relative to n2(a)) we mean a factorization of the form G = G _ diag (rkJ)~=l G+ (2.6) with r(~) = (~ - i)/(~ + i), for ~ E R, kl, k2 E Z, kl _< k2, where the factors G• satisfy the following conditions:

(L+(a)) ~•

for r+(~) = (~ + i)-',

(ii) r_G~_ ' E (L2(R)) 2•

for r_(~) -- (~ - i) -~,

(i) r+G~+' E

(iii) G+IP+GYlI is an operator defined on a dense subspace of (L2(~)): possessing a bounded extension to (L2(R)) 2.

C~mara, dos Samos

411

The generalized factorization is said to be canonical if the partial indices kl, k2 are equal to zero. It is said to be bounded if G+~1 e ( L + ( ~ ) ) 2x~, G_~1 e (L~(a)) 2• The total index of G is given by ind G = kl + k2. Analogous definitions can be given regarding scalar functions g E L ~ (P@ It is well-known that G E ~(L~(P~)) 2x2 admits a generalized factorization (relative to L2(P~)) iff the operator P + O I + : (L+(~)) 2 -~ (L+(P0) ~

(2.7)

(where I + denotes the identity operator on (L~-(I~.))2) is Fredholm. The generalized factorization is canonical iff the operator P + G I + is invertible ([9], [13]). Finally, we will use in this paper the following corona theorem for the algebra

L+~(a): If fl, f2 e L+~(a) and inf Ifl(~)] + If~(~)l > 0, there exist functions fl f2 e ~sC+ L+(R) such that s +/25 = 1 . (zs) An analogous statement holds in the algebra L;c(a) (IS], [10]). 3

EQUIVALENCE

THEOREMS

Let us consider the problem of obtaining conditions for existence and explicit formulas for a generalized factorization of matrix functions of the form bei~

a

G=

(3.1) be-i~

where a, b E L ~ ( ~ ) . Such matrix functions G are diagonalizable and can be represented as a product G = A-1DA (3.2) where 1

e i~

1

--e i~

A =

, D=

dl=a+b,

diag(dl, d2) ,

d2=a-b

.

(3.3)

(3.4)

This problem is related to that of solving an equation of the form Gr + = r (or Gr + = r e - , with r(~) = (~ - i)/(~ + i)) where r177e (L~(P~)) 2. We begin by establishing a relation between equations of this type and analogous ones where G is replaced by a triangular matrix function and which correspond to finite-interval convolution equations. T H E O R E M 3.1 Let G, dl and d2 be defined by (3.1) and (3.4). Let moreover dl and d2 have bounded canonical factorizations dl = dl-dl+, d2 = d2-d2+.

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CLrnara, dos Santos

(i) We have 1_6-1 2-

0

1

e~

(3.5)

T

G =dl 1

1

i"

1

--~6- e- ~

-26+

0

where

6+ = d2+ 6_ = --dl- , dl+ ' d2e -~r

0

t

e i~

(3.6)

,t=6++6_

T=

;

(3.7)

i] (r r is a solution to the Riemann-Hilbert problem Gr + = r r E (L~(P~)) 2, then (r162 is a solution to the Riemann-Hilbert problem T r + = r r 6 (L2i(~)) 2, where r177= (r ~, r are given by

r

= dl+(r + e'~r

r

r

= d-~(e-~r162 + r

= -2d~+r C f = 2d;_~r - .

(ii) If T r + = r with r E (L~(P~)) 2 and T defined by (3.7), where 6+ C (r r is a solution to Gr = r where G is a matrix function of the fo~'m (3.1), given by G = A - l d i a g ( 1 , 6 + 6 -1) A, with A defined by (3.3) and r177= (r162 is given by

L~(R), then

+ 1o,~6-1o1,+ 2~ + ~ '

r162

r PROOF:

6=~r

r162

_1~-1o1,+ 2 ~+ ~

r

1

1

i~

Cf .

(i) The first part can be verified directly and the second part follows

from (3.5). (ii) If Tr + = r

we have

r

= e%~

(6+ + 6_)r

+ e%~+ = r

.

This can also be written in the form 2

5+[(r

+~J

which is equivalent to Gr

= r

9

C~mra, dos Samos

413

It follows that the problem of obtaining conditions for the existence of a generalized factorization for G and explicit formulas for the partial indices and the factors can be reduced to the same problem concerning T. It turns out that this problem can be solved under certain conditions, which do not impose however that 5+, 5_ be invertible in L+(P~), LG(~), respectively. This will be done in the next sections and therefore, if G is such that the corresponding triangular matrix T belongs to the class considered in Section 4, we can use (3.7) to determine a factorization for G. 4

CANONICAL -EXISTENCE

GENERALIZED AND STRUCTURE

FACTORIZATION OF THE FACTORS

OF

T

Let T be a triangular matrix of the form e -i~

0 (4.1)

T= t

e i~

where t E L~(R) is such that, for some a E [0, 1], te - ~

= t+ + t_e -i~ , t+ E L ~ ( ~ )

.

(4.2)

We start by establishing a sufficient conditon for existence of a canonical generalized factorization for T. T H E O R E M 4.1

inf [ei(1-a)~l-{-It+(~)l ~> 0

(4.3)

~e@+ inf le-'~ I + It-(5)l > 0 ,

(4.4)

then T admits a canonical generalized factorization.

PROOF: T admits a canonical generalized factorization iff P + T I + : (L+(R)) 2 -+ (L+(~)) 2 is an invertible operator. Let us first show that it is injective by proving that Tr +=r

r177

2 ,

(4.5)

admits only the trivial solution. This equation is equivalent to

{

e-~%+ = r (4.6)

t+r + + e'(1-~)~r + = e-i"~r

- t_r

and it follows from the second equation in (4.6) that both sides are equal to zero. Thus we have t+r = - e - ~ " ~ r + (4.7)

{

eiO-~}~ = e - ~ r +

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C~mara, dos Santos

and t-r

= e-i~r162

(4.8)

On the other hand, since (4.3) and (4.4) are satisfied, by the corona theorem there are functions t+, E+ C L + ( a ) and t_, E_ E L E ( ~ ) such that t+t+ + ei(1-a)~E+ = 1

(4.9)

e-~
(4.10)

Using the last two equalities, it follows from (4.7) and (4.8) that

~+r

- + =/~-r - t+r

{-

+ t-r

(4.11)

and we see that both sides of (4.11) must be equal to zero. Therefore we have E+r + - {+r

= 0 (4.12)

t+r + + e~(1-~/er + = 0 and, taking (4.9) into account, it follows that r = r = 0 and r = r = 0. Now, let us prove that the operator P + T I + defined above is surjective. This is equivalent to showing that T r + = r + r admits a solution (r r for any r E (L+(R)) 2. However, we can prove instead that

e-%~ = r (4.13) tr + + ~

~=~+r

admits a solution for any u e L + ( ~ ) M E I ( L 2 ( ~ ) ) (cf. [2], [11]). Assuming that r a solution to (4.13) and using the corona theorem as before, we obtain

r

give

eic~{r 1 ~-- J~+r + -- ~+r + "~-~+e--i(1--a){~a+

(4.14)

ei~r162 = t - r

(4.15)

+ E-r

+ t-e~u~

where % = p+(e-i~%). Prom (4.14) and (4.15) we get

p - (ei~{r

= p - ( ~ + e -i(1-~

)

p+(ei~r162-) = p+({_ei~r

(4.16)

(4.17)

and therefore r

= e-'~e [e+ ( L e ~ % a) + P - ({+ e-~(1-~)e~+)]

(4.18)

and r

= e r

(4.19)

= e'(r -

= eia~(t_r

(u s - t+r +)

(4.20)

- u~) .

(4.21)

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415

Having proved that, if (4.13) has a solution, it must satisfy (4.18) - (4.21), we still have to show that r r defined by these equalities, belong to the right spaces L~(~), for any

u 6 L+(R)

fl

g(n[(a)).

Let r

be defined by (4.18). The first term on the right-hand side has the form and it represents a function in L2(R ) because (t_u~) C L[(R); as to the second term, e-{~(P-(t+e-iO-o)(u+), it obviously represents a function in L~-([). Thus, we have r ~ L2 (~). As to r defined by (4.19), we have

e-{~r

r

=

e'(i-a)(p+(~_eia(u; )+ e'(i-a)(P-(e-~O-a)(t+u +) .

The first term on the right-hand side obviously represents a function in L~-(~); as to the second term, it is also in L~-(~) since t'+u + ~ L ~ - ( ~ ) .

Considering now r r

we have

e

-i(1-~){ u s+

--

e-i(1-~){t+ z'+wl

e

-i('-~)( u s+

--

t+p+(~_d~u[~)

--

§ r~-i(*-~)r

--

p+(e-iO-~)r

Using (4.9) we obtain from here r

= -t+P+(Le~"%~)

+ E- + %+ + §~ +D+f~-i(1-a)($ . ~ o +o,+~ ~.j

and it is now clear that r + E L~-(R). Finally, using (4.10), we have

r = t_p+ (f_e~(u~ ) + t_P-(~+e-~(1-~)(u+ ) _ e~(u-~ = - k _ u ~ - t_P-(t_ei~r which shows that r

+ t_P-(t+e -iO-~)r

E L~-(R).

, 9

Sufficient conditions for existence of canonical generalized factorization for a class of triangular matrix functions which reduces to the case studied here (for a = 0 in (4.2)) have been obtained in [3], using a different method which is also related to the corona theorem. The next theorem characterizes what we call the "structure of the factors" in a canonical generalized factorization. T H E O R E M 4.2 If T admits a canonical generalized factorization, such a factorization is given by T = T_T+ where T+

_ei(1-a)r

T;I=

(4.22) T~-

t+

and T1-

_e-ia~

(4.23)

T_=

Tg with T~, T~ such that r•

E L~(~), r•

-t_ E L~(~).

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C~mara, dos Samos

PROOF:

With the assumptions of this theorem, we have

TT+ 1 = T_ ,

(4.24)

the factors T+ 1 and T_ having the form T~

T3+

Tg

T+I=

, T2+

where r•

T~(4.25)

T_=

T+

Tg

Tg

i 6 L~:(P~), j = I, 2, 3, 4 and we can impose a normalizing condition T~-(-i) = O, T ~ ( - i ) # 0 .

(4.26)

To calculate the factors T+ and T_ we solve the homogeneous Riemann-Hilbert

problem Tr +=re-

, r1776 L ~ ( R )

,

(4.27)

where r(~) = (~ - i)/(~ + i) (~ E a) and whose solutions give the columns in T+ 1 and T_ (cf [6]). It should be noted that this is equivalent to the more commonly known non-homogeneous problem Gr + = r + r+uo (Uo C C 2) given e.g. in [9]. Equation (4.27) is equivalent to e - ' % + = Tr

(4.28) tr + + e % + = rr

which, taking (4.2) into consideration, can be put in the form

{

e-%i ~ = rr

(4.29)

t+r + + ei(1-~)~r

= r(e-~r

- t-r

From the second equation in (4.29) it follows that t+r + + ei(1-~)~r + = Kr+ = r ( e - i ~ r

- t_r

(4.30)

for some K C C. The existence of a canonical generalized factorization for T, subject to the normalizing condition (4.27), implies that there is a solution to (4.30) such that r

= 0,

r

# 0

(4.31)

and which corresponds, therefore, to K # 0. On the other hand, (4.30) has an obvious solution, of the form r r which corresponds to K = 0.

= -eiO-~)~r+ , r r

= t+r+ ,

(4.32)

,

(4.33)

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417

This means that, if r conditon (4.31), we have

= ( r 1 ,r

T+

r

= (r162

satisfy (4.27), subject to

T1- _e-iC~(

_ei(1-cJ(

(4.34)

T

T~

t+

-t_

where T+ = r+1r +, T3- = r-l•] - (j = 1, 2). Furthermore, we see from (4.30) that t + T ~ + e~(1-~)~TJ- = K = e - ~ T f

- t_T;-

(4.35)

where K # 0 (and we can assume K = 1). Since the left-hand side of (4.35) represents precisely the determinant of T~+

- e i(1-aM (4.36)

=

Tf t+ while the right-hand side of (4.35) represents the determinant of

_

T1

- e ia~

T~-

-t_

(4.37)

=

it follows that T = T_T+ is a canonical generalized factorization for T, where T+ 1 = T+, T_ = T_. .. It is clear from the proof of Theorem 4.2 that equation (4.30), obtained from (4.27), can be interpreted as a "determinant condition" which gives us information on the structure of the factors T+ I and T_. This idea was presented in [7] for a class of 2 • 2 matrix functions. In this case a "determinant condition" was obtained that showed that one of the columns in the factors could be expressed in terms of the other, thus allowing us to obtain a generalized factorization by just determining one of two columns in the factors. It can be shown that the same procedure can be applied to a rather vast class of matrix functions, including in particular all those mentioned in [6]. In the case of triangular matrices of the type considered in this paper a similar situation occurs, since one of the columns in T+ I and T_ can be known from the start, as expressed in Theorem 4.2. In fact, this type of matrix functions is included in the class considered in the following theorem. T H E O R E M 4.3 Let G be such that it admits a generalized canonical factorization G = G_G+ and det G = 1. I f the equation Gr + = r r can be put in an equivalent form

G1Gr + = r G l r

(r177E L2~(P~))

(4.38)

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C~nara, dos Santos

where G1 is invertible and

GIG

a

b

C+

d+

(4.39)

G1

=

a,b, 5, b 9

C--

c+,d+ 9

d_

c_, d_ 9 L : o ( a )

,

(4.40)

then (apart f r o m a c o n s t a n t m a t r i x f a c t o r ) T+

-d+

G+ 1 =

, T+

T1-

-d_

T 2-

c_

G_ =

c+

(4.41)

with r -+i + T ~ 9 Lf(P~), r - l T j - 9 L ~ ( a ) .

PROOF: If G1 is invertible, it is clear that GIGr + = rGlr

r162Gr + = r e -

.

(4.42)

Thus Gr + = r e - can be put in the equivalent form

a r + br

= r(~ r + ~ r (4.43)

c+r + d§162

= r(c_r

+ d_r

.

An obvious solution to the second equation in (4.43) is given by r

= -d+r+ , r

= c+r+

r

= -d_r_

= c_r_

, r

(4.44) (4.45)

.

As to the first equation in (4.43), we see that when we replace r expressions given in (4.44) and (4.45) we obtain

r

by the

(4.46)

ad+ - bc+ = ~d_ - bc_

which merely expresses the fact that det(G1G) = det G1. Therefore r

= r + [ - d + c+] T , r

= r_[-d_

c_] r

(4.47)

give a non-trivial solution to Gr + = r e - and we can take r+1r +, r - 1 r - as being one of the columns of G+ 1 and G_, respectively. 9 Matrix functions T of the form (4.1) belong to this class not only when t satisfies (4.2) but also, for example, when t belongs to one of the following classes: (i) t = t_t+ + h+e i~ + h _ e -i~

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419

with: t+ e L + ( R ) ,

t ~1 9 L ~ ( ~ ) ,

t -1 9 $ ( L + ( ~ ) )

inf It+(()[ + le*(t-_~(()l > 0 .

In this case, (4.27) is equivalent to

(4.48) (t+ + h+e'~t:l)r + + e ' q - l r + = r ( - t Y _ l h _ r

+ t - l r -) ;

thus the factors in a canonical generalized factorization will be of the form T+

_ei~t--1

T+

t+ + h+e~t -1

,

T1

- t -1

T[

-tiIh_

(4.49)

T_=

A similar result holds if t takes the same form, with t_ E L•(R), t+~1 E L*~(R), t: 1 e

E~(L;~(a)) and i~/~r

+ le-'r

> 0.

The existence of a canonical generalized factorization for matrix functions T with t belonging to this class was established in [3] and a result analogous to (4.49) was obtained, although by a different method. (ii) t is such that for some f_ C L ~ ( R ) N s we have

if_ = t+ + t_e - ~

(4.50)

with t+ E L~(IR). This is a case which includes functions t of the form (4.2) and it follows from Theorem 4.3 that, ff a canonical generalized factorization exists, we have

-f, T+

(4.51)

T_=

t+

T 2-

-t_

since equation (4.27) is now equivalent to e-%t

= rr

(4.52) t + r + + eir162 + = r ( f _ r

- t_r

) 9

As to the existence of a canonical generalized factorization, we can proceed as in the proof of Theorem 4.1, if we impose in addition that inf iei~f_(~)l + It+(~)l > 0 ~eC + inf

]f-(~)l + It-(~)l > 0 .

(4,53) (4.54)

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It is not difficult to see that the conclusions of Theorem 4.3 can be extended to other cases, namely when det G admits a bounded canonical factorization, as we show in an example considered in Section 6. The main feature in all these cases is t h a t equation (4.27) can be put in an equivalent form as a system where one of the equations takes the form of the second equation in (4.43), which can easily be done in all cases considered in this paper. Not only this allows us to know one of the columns in G+ 1 and G_ without further calculations, but it also turns out to be crucial to obtain the other column as a solution to (4.27), as we show in the next section. As a consequence of the previous theorems regarding the structure of the factors in a canonical generalized factorization, we have the following result concerning the behaviour of these factors. T H E O R E M 4.4 If the assumptions of Theorem 4.3 are satisfied and G = G_G+ is a bounded canonical factorization, then inf Ic+(~) I + Id+(~)l > 0 ~e@+

(4.55)

inf Ic_(~)l + Id_(~)l > 0 . ~eC-

(4.56)

PROOF: Since d e t G + 1 = K = d e t G _ , where K is a non-zero constant and G+ 1, G_ take the form (4.41), the conclusion of this theorem follows imediately from the fact that, in this case, T~, T~ 9 L~(R). .. COROLLARY 4.5 If T satisfies the assumptions of Theorem 4.2,

inf lei(1-c~)~[q-It+(~)] > 0

(4.57)

inf le-i~l + [t_(E)[ > 0

(4.58)

are necessary conditions for the factorization to be bounded. W h a t happens if these conditions (which are also sufficient to garante the existence of a canonical generalized factorization) are not satisfied is not completely clear. However, in some cases, we can see t h a t there is no canonical generalized factorization. In what follows we go back to T defined by (4.1) and (4.2). T H E O R E M 4.6 If, for some

Od I

> O, we have

i

(i) t+ = e ~ ~T+, with 7-+ E L+(R) and we have c~ ~ 1 or t

(ii) t_ : e - ~ (T_, with T_ C L~o(R ) and we have c~ # O, T does not admit a canonical generalized factorization.

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421

PROOF: If T = T_T+ is a canonical generalized factorization for T, T+ 1 and T_ take the form (4.22) and (4.23), respectively, where T~, T~ are such that defining r = r• r = r• we have (4.30) with K ~ 0. We will show that, if (i) holds, (4.30) cannot be satisfied with K ~ 0. In fact, we would have

r+eiJ~r + + e~(1-~)~r + = Kr+ . For a0 = min{a', 1 - a}, it follows that 7+ei(J-~o)~r + + ei(1-~-~o)~r + = Kr+e-i~o~ where the left-hand side represents a function in L+(~), while the right-hand side only represents a function in L+(R) if K = 0, since we have a0 > 0. Analogously, if (ii) holds, we have t

e-i~r

- r - e - i ~ r162 = K r _ .

For al = rain {a, a'} if follows that e-i(a-a~)~r

_ T_e-i(a'-aD~r

= Kr_eia~

and we see that such an equality can hold only if K = 0. 5 -

GENERALIZED FACTORIZATION EXPLICIT FORMULAS

.. OF

T

Let the assumptions of Theorem 4.1 be satisfied. In this case, T admits a canonical generalized factorization and it is clear from Theorem 4.2 that we can obtain it explicitly by determining a solution to Gr + = rr

r177= (r

r

9 (L2~(a)) 2 ,

(5.1)

= 0 .

(5.2)

such that r

# 0,

r

In fact, such a solution satisfies (4.30) with K ~ 0 and therefore we can take Tj+ = r=~1r j = 1, 2, in (4.22) and (4.23). We need first the following auxiliary result. LEMMA 5.1 If t+ 9 L+(R) is such that (~.3) is satisfied and c~ ~ 1, there exists )~1 > 0 such that inf

It+({)l > 0 ;

(5.3)

9 (C + + iA1)

f i t _ 9 n ~ ( a ) is such that (~,.~) is satisfied and a ~ O, there exists As > 0 such that inf 9 ( e - - i~:)

It_(~)l > 0 .

(5.4)

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C~nara, dos Santos

PROOF: If (4.3) is satisfied and a ~ 1, we have inf ]ei(1-~)~ I + It+(~)l = c > o ~eC +

(5.5)

On the other hand, there is Ai E [R + such that le~(~-~)9 < 2'

if Im (() > ~

(5.6)

.

If follows from (5.5) and (5.6) that we must have

E<_

inf

It+(~)l +

( ~ (e + + i~) and therefore the first part of Lemma (5.1) is true. We can prove the second part similarly.

R E M A R K 5.2 If we identify t+ with its analytic extension to C + and we denote by the same symbol t+ its restriction to a line R + +iA1, we see that Lemma 5.1 means that t+ E G L + ( R + iA1) and, analogously, t_ E G L ~ ( ~ - iA2). T H E O R E M 5.3 Let (~.3) and (~.4) be satisfied. Let moreover A1 > 0 be such that (5.3) is also satisfied, i r a ~ 1, and A2 > 1 be such that (5.~) holds, if c~ ~ O. A solution to (5.1) - (5.2) is given by (5.7) r

(5.8)

= e-i(1-~)~r+ + t+P~(ei~r+t)

(5.9) r

,

(5.10)

on r2,

for o~E]0,1[ ; (5.11)

= e~r_ + r-~t_P?(e'~rj)

where, for F1 -~ R + iA1 and F2 = R - iA2, we define P~ = P~+ r2-Prl,

{=e-i~t+ 1

P+ = - P P l ' ~ = e-i~t+ 1

P~-=P~,

{ = t _ -1

on

on F2,

on rl, t = t -1

F1,

if

if a = l

c~ =

0

.

;

(5.12) (5.I3)

PROOF: This result can be verified directly. However, it is more useful to prove it in a way which shows how these expressions have been obtained. We must have

{ e-~r + = rr

(5.14) tr + + e%~ + = ref

C~mara, dos Santos

where r

423

= 0, r

# 0. This system is equivalent to

{ e-%~+= ~r (5.15) t+r

+ ~~162

=

r(r e-i~ -

t_r

and from the second equation in (5.15) we see that t+r + + e'(1-a)~r + = K r + = r(r

-'a{ - t_r

(5.16)

for some K ~ 0. We can assume that K = 1, which means that we impose r = ~eia. It follows from (5.16), taking also the first equation of (5.15) into consideration, that rei~r

(5.17)

= r +e-i(1-~)~t +l - r t + 1

Let's denote by F the function defined by the right-hand side of this equation; we remark that F admits an analytic extension to the whole plane C, since e - ' ~ F ( { ) = e-ir162 + = re[

(5.18)

.

Let us assume that ~ E ]0, 1[. From (5.17) we have P - (reinS(91) = P - (r+e-i(1-a)~t+ 1 -- r =

(5.19)

1)

P ~ (r+e-iO-a)~t+ 1) - P ~ (r

1)

(5.20)

(see Lemma 5.4 following the present proof). Now, the second term on the right-hand side of (5.20) is zero, since r 1 9 L+(F~). Therefore, we have P-(re'a~r

= P ~ (r+e-iO-a)~t+ 1) .

(5.21)

On the other hand, it follows from (5.16) that rei~r

= -r+eia~tZ 1 + rr t -1 ,

(5.22)

thus

P+(~"~r

= P+(-~+r

~+ rr

= - P G (r+ei~ft -1) + P ~ (rr t - l ) ,

(5.23) (5.24)

(see Lemma 5.4). Once again, the second term on the right-hand side of (5.24) is zero, for rr 1 E L G ( ~ - iA2). Therefore, P* (~e'~r

= - P+2 (~+e'~t:

1) ,

{5.25/

and, taking (5.25) and (5.2I) into account, we have r

= r - l e - ' ~ f [ P r , (r+e-iO-a)~t+ 1) - P L (r*e~aftT-1)]

(5.26)

r

= e~rr

(5,27) (5.28)

r

= e-~(1-~r

,

- t+r

9

(5.20)

424

C~mara, dos Santos

It is now left to prove that r

r

defined by (5.26)-(5.29), are indeed a solution

to (5.1). Let us first show that re{ 6 L~ (~). We have

rr = e-~
(5.30)

where the first term on the right-hand side obviously represents a function in L~-(R). As to the second term, we see that it represents an entire function, which belongs to L ; (Fz), since r+t_-1 E L~-(F2). Moreover, eicrr = ei(1-aKPFl(e-ir

) -- ei(1-a){P6(eia{r+t -1) ;

(5.31)

the first term on the right-hand side of this identity represents a function in L+(IR), since r+c~ 1 9 L ; ( r h , and the same is obviously true for the second term. Therefore, r 6 L+([R). As to r defined by (5.28), we have

r

= eider- + t-r-l[erl(r+e-i(1-a)r

1) - e~(r+eia{t-_l)]

= t-r-lPr~ (r+e-iO-~)~t+ 1) + t - r - l P ~ (r+ei~t21) .

(5.32) (5.33)

The first term on the right-hand side of (5.33) obviously represents a function in L;(R); the second term represents a function in L;(F2) and, since it can be written in the form

t-r-iP~(r.e'~(t-I)= ei~[r- - t-r-'e-{~(P6(ei~(r+t-1)] ,

(5.34)

where r_t -1 9 L~(F2) and e~ is bounded in the strip 0 > I m (r > -A2, we see that in fact the second term on the right-hand side of (5.33) also represents a function in L;([%). Finally, we can see analogously that r defined by (5.29), belongs to L + (R). The cases where a = 0 and c~ = 1 are analogous, only simpler. 9 LEMMA 5.4 Let F(() denote the right-hand side of (5.17) and let A~, ~2 > 0 be such that (5.3) and (5.~), respectively, are is satisfied. Then

[ F(~)_

[

F(r

VP Ja ~ - - ~ a r = VP Ja VP

F(()d(:Vp a~-z

s

d~, if z 6 C-

F(()d(

-i~r

(5.35)

if z 9 1 6 2

(5.36)

'

PROOF: Let L~ and R~ be the paths in the complex plane represented by

z = - ~ -.t-i y , y 9 [0, /~1] =~-iv

, y9

(5.37) ,

(5.38)

respectively. F being an entire function satisfying (5.18) we have, for z 9 (g-,

VP I. ~-~-~F(~)d~ = alimo~ ( L ,

~-zF(~)d~ +

i~t<,~-]F([)d~ z + VP

i~+ia,

~-zF(~'~) d~ "

(5.39)

Cfimara, dos Samos

425

Now, the first limit on the right-hand side of (5.39) is zero, because F({) = e-i{f+((), with f+ E L~-(~) and e -i~ bounded for ~ in the strip 0 < I m z < A1. In fact,

/oo

{- z

I

- ~ - ~ Y -- zaY[ ~

2

rA~

_
1

I~ + i y - ~I~dy

and since the first integral on the right-hand side represents a bounded [14]), we see that

lim

O~---*q-O0

(5,40)

/ s c~ ~(-~{)zd { = 0

function of a (cf.

(5.41)

and, analogously, f F(~) lira /L d~ = 0 . c, ~ - - Z

t~r

Thus (5.35) holds and (5.36) can be proved in the same way.

(8.42) []

REMARK 5.5 (cf. [2], [12]). The solution to T r + = r e - which is presented in Theorem 5.3 is expressed in terms of the operator PF. In fact, denoting by F the composed contour F = -F1 + F2, the space L2(F) admits a decomposition into a direct sum

&(r) =

Z~-(r) @ L~(P)

(5.43)

where L+(P) = I m P +, L~-(F) = I m ( I - PF) and P+ = 1/2(I + St). Any element ~+ of L+(F) is a function which admits an analytic extension to the strip defined by -A2 < Im (z) < A1, which we also denote by 71+. Thus, when we refer to 77+ as belonging to L2(~), we are referring to its restriction to R, as an analytic function in an open strip containing the real line. It is in this sense that the expression PF(e~a{r+t) must be understood in (5.7) - (5.1o).

In particular, if c~ = 0 and t+~1 e L+(/R), we have

PL (r+e-'q; ') = P- (r+e-'q2)

(5.44)

and thus we obtain r

= eiCp - (e-i(r+t+ 1)

(5.45)

r

= t+P+(e-'{r+t+ 1)

(5.46)

r

= r - l p - (e-i(r+t+ 1)

(5.47)

r

= r_ - t _ r - l P-(e-i{r+t+ 1) .

(5.48)

This will be used later, when we consider an example in section 6. The next result concerns the behaviour of the factors.

426

C~mara, dos Santos

THEOREM 5.6 If conditions (4.3) and (4.4) are satisfied and, for some a 9 [0, 1], t+ 1, t --I can be decomposed in the form t+ 1 = e-~a~(to+ + tl+e i~)

(5.49)

t: ~ = e~fl-~)~(to_ + h_e -~)

(5.50)

where, for some )u >_ 0 such that t+ ~ 9 L + ( R + i,1~), we have to+, t~+ 9 L+($[ + i;~) , to+e -~ 9 L ~ ( N + i)~)

(5.51)

and, for some )~_ > 0 such that t --~ 9 L ~ ( R - i;~:), we have to- , tl-

9 LL(~-

iA~), to_e ~ 9 L + ( e -

i;~) ,

then T admits a bounded canonical factorization, such that T~+, T~_ (cf. (4.22) and (4.23)) can be decomposed in the form T+ = T~,o + T~,~_~ , T~- = T{,o + T~_ ~

(5.52) Theorem 4.2,

(5.53)

with e-~(1-~XT~,o 9 L L ( R ) , e-i(1-~)~T~l_~ 9 L+~(a)

(5.54)

ei~(TZ.o 9 L+(R) , ei~(T~_~ 9 L•(a)

(5.55)

.

Conversely, i f T admits a bounded canonical factorization such that T +, T~- satisfy the above mentioned conditions, then conditions (4.3) and (4.4) hold and t+ ~, t 5~ can be decomposed in the form (5.49) and (5.50) respectively. PROOF: In order to prove the first part of the theorem, we assume that (4.5) and (4.6), as well as (5.49) - (5.52) are satisfied. Then we know by Theorem 4.1 that T admits a canonicM generalized factorization and, by Theorem 4.2, this factorization is bounded iff

where r

r+lr - , T~_1r+ 9 L+($[)

(5.56)

r_-l(~l , r_-1r

(5.57)

9 L~o(~ )

r are given in Theorem 5.3. It is useful to notice first that (5.49) (resp. (5.50)) implies that e-i~eto+ 9 L + ( a + i~1) (resp. ei(1-~)~to_ 9 L L ( a -

i~2)) 9

(5.58)

Moreover, to+ and to- are entire functions and e - ~ t o + 9 L+(a) , e~fl-~)~to_ 9 L ~ ( R ) ,

(5.59)

for e-in'to+ = ei(1-a)~e-i~to+, where e-~to+ 9 L ~ ( R + i)~1) and eifl-~)Q o_ = e-i~ei~to_, with ei~to_ ~ L+(I~ - i;~2).

Cfimara, dos Samos

427

Now we have, for El = [R+ iA~, r2 = [R - iA2:

(5.60) (ei(t0-r+)

(tl-r+) - PF, (e-i(to+T+)

-- PFll (t]+?'+)

.

The second and the fourth terms on the right-hand side of (5.60) are zero (assuming %2 > 1); therefore we have, taking into account that eiqo_ 6 L+(F~) and e-iCto+ 6 L ~ ( F j ,

P?(eia{r+t) = (eiqo_ - K1)r+ - (e-i~to+ - K2)r+

(5.61)

where K1, K2 are constants (K1 : (ei~to-)(-i), K2 = (e-i~to+)(-i) ). Using (5.61) we obtain from (5.7) - (5.10): r~.1r

= e-i=qo+ - ei(1-=)r

- K1 + K~_)

(5.62)

rT)r + = t+(eiqo- + h+ - K1 + K2)

(5.63)

r_-tr - = -e~(1-=)qo- + e-i=~(e-i~to+ - K2 + ]ri'l)

(5.64)

= - t - ( e - i ~ t o - + t l - - K1 +/(2) ,

(5.65)

r-1r

which shows t h a t (5.56) and (5.57) hold. Moreover, since T~ = r+-1 r • we see t h a t T~ can be decomposed in the form (5.53), with T~o = e - i ~ t o + , T~I_,~ = e{O-'~)((e{qo- - Kx + K2)

(5.66)

T(,o = - e i O - ~ ) q o - , TZ,_a = e-i~{(e-iqo+ - K2 + K1) .

(5.67)

Considering now the second part of the theorem, let us assume that T admits a bounded canonical factorization such that T +, T~- satisft (5.53), (5.54) and (5.55). From Theorem 4.4 it follows that (4.3) and (4.4) hold. Furthermore, it follows from the results of Theorem 5.3 that t+ 1 ~___ r+ 1r

+ +~- 1.:.i( ~ 1 - a)(,,."+ I ,s ~'2 = T+ +

t~_lei(1-a)(T+ (5.68)

= T~o + (Tl+,l_a + t+leiO-c~KT +) t -1 = r-_Xr + t l l e - i ~ r i _ l r

= T[ + tlle-i~TZ

=

(5.69) = T~,o + (T~,_a + t-le-ia(T~-) which completes the proof.

9

If, in particular, a = 0 or o~ = 1, the last result can be presented in a simpler form, as we state in the next Corollary. COROLLARY 5.7 (i) For c~ = O, T admits a bounded canonical factorization iff (4.3) and (4.4) are satisfied and

t71 = to+ + tl+ei~

(5.70)

428

C~mara, dos Santos

where to+, tl+ satisfy (5.51). (ii) For ~ = 1, T admits a bounded canonical factorization iff (4.3) and (4.4) are satisfied and t-J 1 = to- + t l - e - ~ where to-, t l - satisfy (5.52).

PROOF: (i) I r a = 0, (5.49) takes the form of (5.70) and (5.50) is always satisfied, with to- - ~i .,nd t l - = t_-1. Thus, if (4.3), (4.4) and (5.70) hold, then T admits a bounded canonical facmrization. Conversely, if T admits a bounded canonical factorization, then (4.3) and (4.4) hold, by Theorem 4.2, and T + E L+(R), T1- E L{o(R ) and e - ~ T + = T 1 so that (see (5.69)) t+ 1 = T + + t+lei~T~

and (5.70) is satisfied, with t + = T +, t + = t + l T +. (ii) Analogous to (i). 6

9

AN EXAMPLE

Let G be a matrix of the form 1

ge i4

ge -i~

1

(6.1) where 1 - p+(1 - p p _ e -ir g = 1 + p+(1 -/3p_e-ir

'

(6.2)

0 < t3 < 1 ,

(~ + 2i~ 1/2 p+(~) = \ - ( ~ - j , p-(~) = ( ~ - i) -'/2 , p-~ e c ( ~ ) ,

p+(oo) = I .

(6.3)

In this case, the eigenvalues dl and d2 are given by dl = 1 + g =

2 1 + p+(1 - ~p_e -~,')

d2=l-g=

l+p+(1-~p_e-ir

2p+(1 - Zp_e-'~)

(6.4) '

(6.5)

We can see that dl and d2 admit canonical bounded factorizations since dl, d2 E C " ( ~ ) and inddl = indd2 = 0. So, let dl = dl-dl+, d2 = d2-d2+ be canonical factorizations of dl and d2. By Theorem 3.1, the problem of determining a bounded factorization for U is reduced to a similar problem relative to 0

T=

)

(6.6)

CLmara, dos Santos

429

where d2+ dlt = ~ + ~ = p+ + (1 -

l~p_e-'r -'

(6.7)

We see from (6.7) that t is of the form (4.2), for ~ -- 0 and t+ = p+ + 1,

t_ = / ~ p _ ( 1 -

~p_e-'r -'

(6.8)

In this case conditions (4.3) and (4.4) are satisfied, so that T (and G) admits a canonical generalized factorization T = T_T+ by Theorem 4.1 and we have T+

_eir

T+'=

T 1,

T+

-I

T_=

-/3p_

p+ + 1

T2-

(6.9)

1 - #~p_e-i~

Since t# 1 9 L+(a), we can use formulas (5.45)- (5.48) to determine and, by means of (3.5), we get a canonical bounded factorization for G, given by 1 i~- 1 ~-e (p;. -- 1)

1 1 ~:'e'{[g_ + ~(p: + 1)g+]

(6.1o)

G+ 1 = did 1 1 --~(p~_ -I- 1)r~lff+ 1 (1 -

T~, T~

1

-~(p:

1

+ 1)

_1r

1 1g_ ~p_e -i{) + -~p_r+

(6.11)

G_ = dl1

i

1

i~-

r+~(1 - ~p_e- ~)g_ - ~e- (1 - ap_e -~)

-1 + -~Ip_e-

where g• = P• It may be interesting to note that the generalized factorization of G could be obtained without resorting to T, by studying the equation Gr + = re- directly. In fact, this equation is equivalent to d1+(r + + e'~r +) = rd[l(r

+ eir162)

d2+(r + - eir162 +) = rd2](r

- ei~r

(6.12) .

Due to the form of dl and d2 in this case, by adding the left-hand sides and the right-hand sides of these two equations, we can write (6.12) in the equivalent form d1+(r + + P~r

= rd~l(r

+ e~r (6.13)

d]+(D]+r + + D2+r +) = rdrl(Dl-r

+ D2-r

where DI+=I+p+ D 1 - = 1 + (1 -

, D2+=(1-p+)e

~p_e-'r -'

~

, D 2 - = - ~ p _ ( 1 - / ~ p _ e - i ~ ) -z

(6.14) (6.15)

C~mara, dos Santos

430

We see therefore that G satisfies all the assumptions of Theorem 4.3, except for the value of its determinant (det G = did2). However, since det G admits a bounded canonical factorization det G = d_d+ with

d_ = dl-d2- = d e t G _ , d+ = dl+d2+ = d e t G + ,

(6.16)

it is easy to see that we can extend the results of Theorem4.3 to this case, taking d+ and d_ into account. From the second equation in (6.13) we obtain d1+(D1+4 + + / ) 2 + 4 +) -- K r + --- rdTl(Dl_4~ + D2-42) .

(6.17)

Considering that we no longer have det G = 1 but, instead, we have det G = d_d+, we rewrite (6.17) as a "determinant condition" of the form

{ d+Idl+(Dl+4[ + D2+4 +) = Kr+d+ ~ (6.18)

d_d[J(n,_4~ + D~-4E) = Kr_d_ and we can see, as in the proof of Theorem 4.3, that

r

= -d+ldl+D2+r+ = -d2+lD2+r+,

4r = -d_d~JD2_r_ = -d2-D2_r_ ,

4 + = d+ldl+Dl+r+ = d21+D~+r+ (6.19)

42 = d-d~-D~- r- = d2_D~_r_

(6.20)

give a solution to (6.13), for which K = 0 in (6.17). Thus we have

g+

-d2~D2+

g[

-d2-D2-

G~ 1

(6.21) 9+

d2~_D1+

g~

d2-D1-

and we see that we can obtain one of the columns directly from the second equation in (6.13), while the other must be obtained by solving the equation G4 + --- r4- subject to some condition which corresponds to K ~ 0 in (6.17). In this case we can impose, for

instance, 4~(--i) = 0, 4 ~ ( - i ) r 0. 7

APPLICATION

TO A CORONA PROBLEM

As a consequence of the previous results we get an interesting byproduct related to the corona theorem. This theorem was used to prove the existence of a canonical generalized factorization for T (Theorem 4.1). It is clear from the second part of that proof that we can obtain the solutions to Gr + = r4- , and therefore the factors G_ and G+, if the functions t+, E+, t_, E_ in (4.9) and (4.10) are known. However, this does not happen in general and we can use instead the canonical factorization of G to determine those functions. In fact, if 2 x 2 matrix function G admits a bounded canonical factorization, its determinant also admits a (scalar) bounded canonical factorization det G = d_d+. Given one of the columns in G+ I and G_, for instance the second, the problem of determining the

C,~mara, dos Santos

431

other column can be put as follows: "Determine 9 +, 9 + 9 L+(P~), 9~-, g2 9 L~(R) such that g+9~+ +-+ + g2 g2 = d+ 1 g~-9~- + g~9~ = d_ , where [g+ g~]T and [g[- g~-lT are the second columns in G+ 1 and G_, respectively". Since this problem can be reduced to the case where d+ 1 = 1, we see that it corresponds to two corona problems, in L+(P~) and L~(~) respectively. In the case of a matrix function G satisfying the assumptions of Theorem 4.3, one of the columns in G+ 1 and G_ is known (see (4.41)). Therefore, if the factorization is bounded, the first column in G+ 1 gives a solution to the following corona problem: "Given c+, d+ E L+(~), determine ~+, d+ 6 L+(P~) such that c+fi+ + d+d+ = 1". More precisely, we obtain the solution to that corona problem which also satisfies a condition corresponding to the first equality in (4.43) for r = r+~+, r = r+d+. An analogous corona problem concerning c_, d_ C L~(P~) can be solved using the first column in G_ (see (4.41)). Considering, for example, a matrix T of the form (4.1) with t satisfying (4.2), we see that {+ = r+1r +, E+ = r+1r " (where r and r are defined in Theorem 5.3) give a solution to the corona problem (4.9), t+t+ + E+/~+ = 1 with corona data t+, E+ E L+(R) (E+(() = d (1-")~, ~ 6 [0, 1]) if the factorization is bounded. Theorem 5.6 gives conditions for this to happen. If the factorization is not bounded we conclude that there is no solution to the corona problem (4.9) such that e-i(t+ 6 L~(R) - - although a solution exists in L+(P~), if (4.3) is satisfied. Analogous conclusions can be reached concerning the corona problem in L~ (P~)

(see (4.10))

t_L + E_L

= i

with corona data t_, E_ 9 L~(P~) (E_(~) -- e -'a~, a 9 [0, 1]).

REFERENCES [1] ABLOWITZ, M. J., CLARKSON, P. A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering; London Mathematical Society Lecture Note Series; 149, Cambridge University Press, 1992. [2] BASTOS, M. A., dos SANTOS, A. F.: Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility; Int.Eq.Op.Th., 16 (1993) 186-223. [3] BASTOS, M.A., KARLOVICH, Yu.I., dos SANTOS, A.F., TISHIN, P.M.: The corona theorem and the existence of canonical factorization of triangular AP-matrix functions; J. Math. An. Appl., 223 (1998) 494-522. [4] BASTOS, M.A., KARLOVICH, Yu.I., dos SANTOS, A.F., TISHIN, P.M.: The corona theorem and the canonical factorization of triangular AP-matrix functions - effective criteria and explicit formulas; J. Math. An. Appl., 223 (1998) 523-550.

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Cgmara, dos Santos

[5] BASTOS, MA., KARLOVICH, YuI., dos SANTOS, AF.: Oscillatory Riemann-Hilbert Problems and the Corona Theorem; Preprint. [6] CAMARA, MC., dos SANTOS, AF.: A non-linear approach to generalized factorization of matrix functions; Op. Th. Adv. Appl., 102 (1998) 21-37. [7] CAMARA, M. C., dos SANTOS, A. F.: Wiener-Hopf factorization of a generalized Daniele-Khrapkov class of 2 • 2 matrix symbols; Math. Meth. Appl. Sc., 22 (1999) (to be published). [8] CARLESON, L.: Interpolations by bounded analytic functions and the corona problem; Ann. Math., 76 (1962) 547-559. [9] CLANCEY, K., GOHBERG, I.: Factorization of Matrix Functions and Singular Integral Operators; Birkhguser, 1981. [10] DUREN, P.: Theory of H p Spaces; Academic Press, 1970. [11] KUIJPER, A., SPITKOVSKI: On convolution equations with semi-almost periodic symbols on a finite interval; Int, Eq. Op. Th. 16 (1993) 530-538. [12] LOPES, P. A., dos SANTOS, A. F.: A new approach to the convolution operator on a finite interval; Int. Eq. Op. Th., 26 (1996) 460-475. [13] MIKHLIN, S., PROSSDORF, S.: Singular Integral Operators; Springer, 1986. [14] RIESZ, M.: A Sur les fonctions conjugu~es; Math. Z., 27 (1927) 218-244. Departamento de Matems Instituto Superior T~cnico Av. Rovisco Pais 1049- 001 Lisboa Portugal 1991 Mathematics Subject Classification: 45E10, 47B35 Submitted: January 10, 1999 Revised: June 29, 1999