The Corona Theorem and the Existence of Canonical Factorization of Triangular AP-Matrix Functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

223, 494]522 Ž1998.

AY985968

The Corona Theorem and the Existence of Canonical Factorization of Triangular AP-Matrix Functions* M. A. Bastos, Yu. I. Karlovich, A. F. dos Santos Departamento de Matematica, Instituto Superior Tecnico, A¨ . Ro¨ isco Pais, ´ ´ 1096 Lisboa, Portugal

and P. M. Tishin Hydroacoustic Department, Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Preobrazhenskaya Str. 3, 270100 Odessa, Ukraine Submitted by Joseph A. Ball Received August 4, 1997

Using the corona theorem, canonical factorizability criteria for triangular almost ilj 0 x e yi l j are obtained. This allows

periodic matrix-valued functions of the form w feŽ j .

the authors to obtain effective criteria for the existence of canonical factorization as well as to construct explicit factorizations for new classes of almost periodic matrix functions in a companion paper. Q 1998 Academic Press Key Words: canonical factorization; corona theorem; almost periodic matrix function

1. INTRODUCTION The problem of factorization of bounded measurable matrix-valued functions of the form GŽ j . s

e i lj fŽ j .

0 eyi lj

Ž 1.1.

* Work sponsored by F.C.T. ŽPortugal. under grants Praxis XXIr2r2.1rMATr441r94 and Praxis XXIrBCCr4355r94. 494 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

THE CORONA THEOREM

495

has received considerable attention in the literature as it is related to the Fredholm and invertibility study of classes of convolution type operators on a finite interval w24, 15]17, 3, 4x. Many authors have until now studied this problem, w17, 19, 20, 13, 2, 9, 25x. In this paper, using the corona theorem, the factorization of a new class of matrix-valued functions is studied that gives criteria for canonical factorization and gives rise to general formulas for the factors. This allows the definition of effective criteria for canonical APW factorization as well as the calculation of the factors for new classes of APW matrix functions, i.e., almost periodic ŽAP. matrix functions of the Wiener class. The actual calculation of the factors, which gives a meaningful application of the results obtained in this paper, is left to a companion paper w1x. This paper is organized as follows. In Section 2 we recall the notions of canonical generalized factorization and of AP factorization that we need. In Section 3 we give a sufficient condition for the existence of canonical generalized factorization of a class of bounded measurable matrix functions of the form Ž1.1.. Section 4 deals with the same question for AP factorization. In general it is necessary to solve a corona problem in order to obtain the factors G " of a canonical factorization G s Gq Gy. However, if G is AP factorizable, with additional conditions, without solving a corona problem it is possible to obtain an explicit expression for dŽ G . s M Ž Gq . M Ž Gy ., where M denotes the Bohr mean value. In Section 5 criteria for canonical AP factorization are given for the class of APW matrix-valued functions Ž1.1.. Formulas for the factors in terms of corona solutions are also given. To obtain these criteria we use an algebraic automorphism of the nonclosed algebra of APW matrix functions of bounded spectrum. In this section we state also a result about the solvability of the corona problem for APW functions of bounded spectrum. 2. PRELIMINARIES To begin with, we introduce the definitions of the spaces and the different notions of factorization that we use in the sequel. In all that follows AP will denote the smallest subalgebra of L`ŽR., containing the functions el: R ª C, x ¬ e i l x , l g R. If f g AP, the Bohr mean value is the number M Ž f . s lim

1

T

H f Ž x . dx Tª` 2T yT

and the Fourier spectrum of f is the set V Ž f . s  l g R: fˆŽ l . s M Ž eyl f . / 0 4 .

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Every f g AP can be expressed as a formal Fourier series Ý lg V Ž f . fˆŽ l. el and we say that f g APW if Ý l g V Ž f . < fˆŽ l.< - `. AP " and APW" are the subalgebras of AP and APW such that AP "s  f g AP: fˆŽ l . s 0 for "l - 0 4 ,

APW" s AP "l APW .

A Lebesgue measurable function w: R ª w0, `x is called a weight if the set wy1 Ž 0, `4. has measure zero. We denote by L p ŽR., 1 F p F `, the usual Lebesgue spaces on R and, given a weight w on R, we define L p ŽR, w ., 1 - p - `, as the weighted Lebesgue space with the norm 1rp

5 f 5 p, w s

ž

HR< f Ž x . <

p

w p Ž x . dx

/

.

The operator SR defined on L p ŽR, w . by

Ž SR f . Ž t . s lim

f Ž x.

1

t g R,

dx, H «ª0 p i R_J Ž t . x y t

Ž 2.1.

«

where J« Ž t . sx t y « , t q « w, is bounded if and only if w11x sup J

ž

1
1rp

w p Ž x . dx

HJ

/ ž

1
1rq

wyq Ž x . dx

HJ

/

- `,

Ž 2.2.

where J runs through all intervals of R, < J < denotes the length of J, and py1 q qy1 s 1. The set of all weights w on R satisfying Ž2.2. is the set of Muckenhoupt weights, usually denoted by A p ŽR.. Assume 1 - p - ` and w g A p ŽR.. Then SR2 s I, which implies that the operators P "s Ž I " SR . r2 are bounded complementary projections on L p ŽR, w .. Let us define the weighted Hardy spaces on the half-planes P "s  z: "Im z ) 04 . H`" are the Hardy spaces of bounded analytic functions on P ". Let qŽ . Ž . Ž Ž .. Hp w s Pq Ž L p ŽR, w .. and Hy fg p w s Py L p R, w . Functions "Ž . Hp w can be extended to analytic functions on P ", respectively. We identify the elements of H`" and Hp" Ž w . with their nontangential limits on R. Before giving the definitions of the different notions of factorization that we will use in the sequel, let us note that, for an n = m matrix function, we say that this function belongs to some space w X x n, m if each entry belongs to X. If m s 1 we write w X x n .

497

THE CORONA THEOREM

Let G be a complex matrix function in w L`ŽR.x n, n and let us recall the definitions of canonical generalized factorization relative to L p ŽR, w . and of bounded canonical generalized factorization Žsee, e.g., w23, 22, 27, 28x.. DEFINITION 2.1. A matrix-valued function G g w L`ŽR.x n, n is said to admit a canonical generalized factorization relative to L p ŽR, w ., 1 - p - `, w g A p ŽR., if G has a representation of the form G s Gq Gy ,

Ž 2.3.

where y1 Ži. rq G q g w H q Ž .x Ž y 1 .x n , n , r y G y g rq G q g w Hq p w n, n , q w y1 .x y1 yŽ .x w Hy Ž w Ž . Ž .y1 ; q w n, n , ry Gy g H p w n, n , and r " x s x " i y1 y1 Žii. the operator Gy Py Gq I possesses a bounded extension to w L p ŽR, w .x n .

Using the passage to the unit circle G with the help of the linear fractional transform

a Ž x. s

xyi xqi

,

x g R,

we get from Theorem 2.2 in w22x that a canonical factorization Ž2.3. only with the property Ži. is determined uniquely to within a constant invertible matrix multiplier. Applying the transform a , from an analogue of Simonenko’s results w27, 28x, Žsee w5, Theorem 6.32x and also w10, Chap. 8.3, Theorem 3.1x. we obtain the following criterion. THEOREM 2.2. The operator TG s Pqq GPy with coefficient G g w L`ŽR.x n, n is in¨ ertible on the space w L p ŽR, w .x n , 1 - p - `, w g A p ŽR. if and only if G admits a canonical generalized factorization relati¨ e to L p ŽR, w .. DEFINITION 2.3. A matrix-valued function G g w L`ŽR.x n, n is said to possess a bounded canonical generalized factorization if G has a representation of the form Ž2.3., where "1 Gq g H`q

n, n ,

"1 Gy g H`y

n, n .

Finally, we give the definition of AP factorization Žsee w15, 17x.. DEFINITION 2.4. A matrix-valued function G g wAPx n, n is said to possess an AP factorization if it can be represented as a product G s Gq LGy ,

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BASTOS ET AL.

"1 "1 where Gq g wAPqx n, n , Gy g wAPyx n, n , and

L s diag w el1 , . . . , el n x ,

l j g R.

The exponents l j are called the partial AP indices of G. If such a factorization exists the tuple  l1 , . . . , l n4 is unique, and if all the partial AP indices equal zero the AP factorization is called canonical and the matrix d Ž G . s M Ž Gq . M Ž Gy . with M Ž G " . defined elementwise is unique w17x. "1 qx The AP factorization is an APW factorization if Gq g wAPW n, n and "1 yx w Gy g APW n, n . The importance of the AP factorization and the role of the matrix dŽ G . for the Fredholm theory of singular integral operators with semi-almost periodic coefficients and of convolution type operators with semi-almost periodic symbols on semi-axes andror on systems of finite intervals are shown in w17, 14, 6, 18, 12x. The following theorem illustrates the role of AP factorization in case of almost periodic coefficients or symbols. On the space w L p ŽR.x n , 1 - p - `, consider the singular integral operator TG s Pqq GPy and the corresponding convolution type operator Fxq I, WG s xy I q F y1 GF where G g wAPW x n, n , x " are the characteristic functions of the semi-axes R ", and F is the Fourier transform. THEOREM 2.5 Žw12x, see also w14, 29x.. ing assertions are equi¨ alent: Ž1. Ž2. Ž3. Ž4. Ž5. Ž6.

If G g wAPW x n, n , then the follow-

the operator TG is Fredholm on w L p ŽR.x n ; the operator TG is in¨ ertible on w L p ŽR.x n ; the operator WG is Fredholm on w L p ŽR.x n ; the operator WG is in¨ ertible on w L p ŽR.x n ; G admits a canonical AP factorization; G admits a canonical APW factorization.

Since the corona theorem plays a fundamental role in this paper, we recall its statement below Žsee w7, 8x..

499

THE CORONA THEOREM

THE CORONA THEOREM. inf

Let f 1 , f 2 , . . . , f n be functions in H`q such that < f j Ž z . < ) 0.

max

zgP q js1, . . . , n

Ž 2.4.

Then there exist functions f˜1 , f˜2 , . . . , f˜n in H`q such that n

Ý f j f˜j s 1.

Ž 2.5.

js1

We refer to f 1 , f 2 , . . . , f n as corona data and to the problem of determining functions f˜1 , f˜2 , . . . , f˜n as the corona problem; f˜1 , f˜2 , . . . , f˜n will be referred to as corona solutions. Note that condition Ž2.4. is also necessary for the solvability of the corona problem Ž2.5..

3. GENERALIZED FACTORIZATION AND THE CORONA THEOREM Let us consider the matrix function GŽ j . s

e i lj fŽ j .

0 yi lj

e

,

j g R,

Ž 3.1.

where l ) 0 and f Ž j . s f 1 Ž j . y hq Ž j . e i lj y hy Ž j . eyi lj

Ž 3.2.

with f 1 g L`ŽR., h "g H`" . THEOREM 3.1.

Let G be defined by Ž3.1. and Ž3.2., and f 1 Ž j . s fq Ž j . fy Ž j . ,

where either Ži.

y1 fqg H`q , fy" 1 g H`y , el fy g H`q , and y1 inf max  < fq Ž z . < , < e i l z fy Ž z . <4 ) 0

zgP q

or Žii.

y1 fyg H`y , fq" 1 g H`q , ey l fq g H`y , and y1 inf max  < fy Ž z . < , < eyi l z fq Ž z . < 4 ) 0.

zgP y

Ž 3.3.

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BASTOS ET AL.

Then the matrix function G admits a canonical generalized factorization relati¨ e to L p ŽR, w . for each p gx1, `w and each w g A p ŽR.. Proof. Let G1 Ž j . s

e i lj f 1Ž j .

0 yi lj

e

s

1 hq Ž j .

0 1 GŽ j . 1 hy Ž j .

0 . Ž 3.4. 1

Hence the matrix functions G and G1 can have a canonical generalized factorization only simultaneously. Let us show that G1 with condition Ži. has a canonical generalized factorization. By the corona theorem, for the corona data fq, gqg H`q , where y1 gq Ž j . s e i lj fy Ž j .,

Ž 3.5.

there exist corona solutions f˜q, ˜ gqg H`q such that fq f˜qq gq ˜ gqs 1. Let Aqs

fq

ygq

˜gq

f˜q

Ays

,

0 yfy

fy1 y . 0

Ž 3.6.

"1 "1 Then Aq g w H`qx 2, 2 and Ay g w H`yx 2, 2 . The matrix function G1 has a canonical generalized factorization if and only if the same is true for the matrix function G 2 defined by

1 G 2 s Aq G1 Ays g 2

0 1 ,

Ž 3.7.

where g 2 s yf˜q fy ey l g L`ŽR.. By Theorem 2.2, the matrix function G 2 admits a canonical generalized factorization relative to L p ŽR, w . for each p gx1, `w, w g A p ŽR. because the operator I g 2 Py

0 I

is invertible on every space w L p ŽR, w .x 2 , 1 - p - `, w g A p ŽR.. Thus the matrix function G with condition Ži. of Theorem 3.1 has a canonical generalized factorization. Let us obtain the factors. Consider y G 2 Ž j . s Gq 2 Ž j . G2 Ž j . ,

j g R,

with G 2" Ž j . s

1 g 2" Ž j .

0 1

Ž 3.8.

501

THE CORONA THEOREM

and g 2 Ž j . s yf˜q Ž j . fy Ž j . eyi lj .

g 2" s ry1 q P " rq g 2 ,

Ž 3.9.

We shall show that G 2" are factors of a generalized factorization. Since rq g 2 g L`ŽR. and w g L p ŽR., wy1 g L q ŽR. for each w g A p ŽR., we conclude that rq g 2 g L p ŽR, w . l L q ŽR, wy1 .. y1 Then taking into account that ry rq g H`y we have q q y1 rq gq ., 2 s Pq rq g 2 g H p Ž w . l Hq Ž w y1 y y y1 ry gy ., 2 s ry rq Py rq g 2 g H p Ž w . l Hq Ž w

and the factors Ž3.8. satisfy condition Ži. of Definition 2.1 of a canonical generalized factorization relative to L p ŽR, w .. Since G 2 admits a canonical generalized factorization relative to L p ŽR, w ., 1 - p - `, w g A p ŽR. and a canonical factorization with condition Ži. is determined uniquely to within a constant invertible matrix, we conclude that Ž3.8. are factors of a canonical generalized factorization of G 2 relative to L p ŽR, w .. Finally, from Ž3.7. and Ž3.8. we obtain the factors of a canonical generalized factorization of G relative to L p ŽR, w ., 1 - p - `, w g A p ŽR., 1 Gqs yh q

0 1

f˜q

gq

yg˜q

fq

1 gq 2

0 1

and 1 Gys gy 2

0 1

yfy1 y 0

0 fy

1 yhy

0 1 .

If condition Žii. is satisfied we also obtain the assertion of the theorem with the help of the transformation G ª JG*J, where J s w 01 10 x and G* is the conjugate transpose matrix for G. The proof of Theorem 3.1 gives also formulas for the factors G " of canonical generalized factorization G s Gq Gy. COROLLARY 3.2. Under the conditions of Theorem 3.1, in case Ži., 1 Gqs yh q Gys

0 1

f˜qq gq gq 2 yg˜qq

0

yfy1 y

fy

y1 ygy 2 fy

fq gq 2

1 yhy

gq fq 0 1 ,

,

Ž 3.10.

Ž 3.11.

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BASTOS ET AL.

˜ gqg H`q are corona solutions for the where gqs el fy1 y , the functions fq, ˜ q corona data fq, gqg H` , and ˜ g 2" s yry1 q P " rq fq fy ey l ;

Ž 3.12.

q yfy1 q g2 fq

Ž 3.13.

in case Žii., 1 Gqs yh q Gys

0 1 fy

yg˜yq

yfy1 q , 0

gy gy 2 fy

f˜yq

1 yhy

gy 2 gy

0 1 ,

Ž 3.14.

˜ gyg H`y are solutions for the corona where gys ey l fy1 q , the functions fy, ˜ y data fy, gyg H` , and ˜ g 2" s yry1 q P " rq fy fq el .

Ž 3.15.

Note that under the conditions of Theorem 3.1 the factors f " in Ž3.3. belong to L`ŽR., but may not belong to AP. For example, let l s 2, fqg H`q , and fy Ž j . s Ž 1 y ceyi js Ž j . .

y1

j g R,

,

where

s Ž z. s 2i

z

H0

sin t t

dt,

z g C,

and c - 5 ey1 s 5y1 H`y . Let us remark that Žsee w24x. s Ž j . s p i sgn j q O Ž< j
Let G be defined by Ž3.1. and f Ž j . s fq Ž j . fy Ž j . ,

where fq" 1 g H`q ,

y1 ey l fq g H`y ,

fy" 1 g H`y ,

y1 el fy g H`q .

Then the matrix function G admits the bounded canonical generalized factorization G s Gq Gy

503

THE CORONA THEOREM

with Gq Ž j . s Gy Ž j . s

fy1 q Žj .

e i lj fy1 y Žj .

0

fq Ž j .

0

yfy1 y Žj .

fy Ž j .

eyi lj fy1 q Žj .

,

Ž 3.16.

.

Ž 3.17.

"1 "1 Proof. Obviously, Gq g w H`q x2, 2 , Gy g w H`y x2, 2 , and G s Gq Gy.

To illustrate how this corollary follows from the proof of Theorem 3.1, put h "s 0,

y1 gq Ž j . s e i lj fy Ž j .,

gy Ž j . s eyi lj fy1 q Žj .

and y1 f˜qs fq ,

˜gqs 0.

Then y1 g 2 Ž j . s yfy Ž j . eyi lj fq Ž j . s yfy Ž j . gy Ž j .

and gq 2 s 0,

gy 2 s yfy gy ,

because fy gyg H`y . It remains only to substitute all these functions in Ž3.10. and Ž3.11..

4. SUFFICIENT CONDITIONS OF CANONICAL AP FACTORIZATION We begin by relating a canonical AP factorization and a canonical generalized factorization. PROPOSITION 4.1. Let G g w AP x n, n admit a canonical AP factorization G s Gq Gy and a canonical generalized factorization relati¨ e to L p ŽR, w ., G s Gq Gy, where 1 - p - `, w g A p ŽR.. Then Gqs Gq Cy1 and Gys Gy, where C is an in¨ ertible constant complex matrix. CG Since AP "; H`" Žsee the proof of Theorem 1 in w21, Chap. 6x. and hence, a canonical AP factorization is a case of a canonical generalized factorization, Proposition 4.1 follows from the uniqueness of canonical generalized factorization to within a constant invertible complex matrix.

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THEOREM 4.2. Let G g w AP x 2, 2 be defined by Ž3.1. and Ž3.2. with h "g AP " and f 1 such that f 1 Ž j . s fq Ž j . fy Ž j . , where either Ži.

"1 y y1 q fqg APq W , fy g APW , el fy g APW , and y1 inf max  < fq Ž z . < , < e i l z fy Ž z . <4 ) 0

zgP q

or Žii.

"1 q y1 y fyg APy W , fq g APW , ey l fq g APW , and y1 inf max  < fy Ž z . < , < eyi l z fq Ž z . < 4 ) 0.

zgP y

Then the matrix function G admits a canonical AP factorization G s Gq Gy with factors G " gi¨ en by Ž3.10. and Ž3.11. in case Ži. and by Ž3.13. and Ž3.14. in case Žii.. Proof. For the proof we consider only condition Ži. Žthe proof with condition Žii. is analogous.. Since AP "; H`" , from Theorem 3.1 it follows that the matrix function G admits a canonical generalized factorization relative to L p ŽR, w ., 1 - p - `, w g A p ŽR., G s Gq Gy ,

Ž 4.1.

where G " are defined by Ž3.10. and Ž3.11. with f˜q, ˜ gqg H`q and g 2" given by Ž3.12.. Let G1 be defined by Ž3.1. and Ž3.2. with h "s 0. By Theorem 2.2, the operators TG s Pqq GPy and TG 1 s Pqq G1 Py are invertible on each space w L p ŽR, w .x 2 . Taking into account that G1 g APW we get from Theorem 2.5 that the matrix-valued function G1 admits a canonical AP factorization y G1 s Gq 1 G1 , qx yx . " 1 g wAPW Ž y. " 1 g wAPW where Ž Gq 1 2, 2 and G 1 2, 2 . Then we obtain an AP factorization of the matrix-valued function G,

G s Gq Gy ,

Ž 4.2.

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THE CORONA THEOREM

where 1 Gqs yh q

0 q 1 G1 ,

1 Gys Gy 1 yh y

0 1

"1 "1 and Gq g wAPqx 2, 2 , Gy g wAPyx 2, 2 . From Ž4.1., Ž4.2., and Proposition 4.1 we may conclude that a canonical generalized factorization of G is a canonical AP factorization. Moreover, to within an invertible constant matrix multiple C, the factors G1" g wAPW"x 2, 2 are given in view of Ž3.10. and Ž3.11. by

Gq 1 s

Gy 1 s

f˜qq gq gq 2 yg˜qq

fq gq 2

0

yfy1 y

fy

y1 ygy 2 fy

gq fq

,

.

Ž 4.3.

Ž 4.4.

Notice that Theorem 4.2 gives conditions for canonical APW factorization of G in case h "g APW" . Let us also note that, under condition Ži. in the last theorem, the factors of a canonical AP factorization of G are defined by Ž3.10. and Ž3.11., q where f˜q, ˜ gqg H`q are corona solutions for the corona data fq, gqg APW " y1 and g 2 s yrq P " rq f˜q fy ey l . Since from Ž4.3., q f˜qq gq gq 2 g APW ,

q ˜gqy fq gq 2 g APW ,

and

Ž f˜qq gq gq2 . fqq Ž ˜gqy fq gq2 . gqs f˜q fqq ˜gq gqs 1, q the functions f˜qq gq gq gqy fq gq 2 and ˜ 2 are corona solutions in APW for q q the corona data fq, gqg APW . Thus for the corona data fq, gqg APW , q Ž there exist corona solutions in APW for the solvability of the corona problem in the class APq; H`q , see w30x.. More precisely, proving Theorem 4.2, we have, in addition, the following accessory result.

Ž . " 1 g APy COROLLARY 4.3. For corona data fq, gqg APq W with ey l gq W q for some l G 0, the corona problem is sol¨ able in APW if and only if inf max  < fq Ž z . < , < gq Ž z . < 4 ) 0.

zgP q

Ž 4.5.

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BASTOS ET AL.

q Moreover, if we have corona solutions f˜q, ˜ gqg APW , then g 2 s q q ˜ yfq fy ey l g APW and by Ž4.3. and Ž4.4., the functions fq g 2 , gq gq 2 g APW y y1 y and g 2 f y g APW . Hence, q q ˜ gq gq Ž gq gq 2 s fq Ž fq g 2 . q ˜ 2 . g APW , y y1 y gy 2 s fy Ž g 2 f y . g APW .

. Ž y1 . Ž . In addition we have M Ž gy 2 s M rq Py rq g 2 s 0 and then 3.10 and Ž3.11. give dŽ G. s

1 yM Ž hq .

0 1

d11 d 21

d12 d 22

1 yM Ž hy .

0 , 1

where d11 s M Ž gq . M Ž fy . ,

d12 s y M Ž f˜q . q M Ž gq . M Ž g 2 . rM Ž fy . ,

d 21 s M Ž fq . M Ž fy . ,

d 22 s M Ž ˜ gq . y M Ž fq . M Ž g 2 . rM Ž fy . .

On the other hand if G satisfies condition Žii., the factors are defined by Ž3.13. and Ž3.14., where f˜y, ˜ gyg H`y are corona solutions for the corona y Ž y. data fy, gyg APW ; H` , and g 2" are given by Ž3.15.. y Analogously we may have corona solutions f˜y, ˜ gyg APW and then dŽ G. s

1 yM Ž hq .

0 1

d11 d 21

d12 d 22

1 yM Ž hy .

0 , 1

where d11 s M Ž ˜ gy . y M Ž g 2 . M Ž fy . rM Ž fq . , d12 s y M Ž f˜y . q M Ž g 2 . M Ž gy . rM Ž fq . , d 21 s M Ž fq . M Ž fy . ,

d 22 s M Ž fq . M Ž gy . , and

g 2 s yf˜y fq el .

Let us indicate a generalization of Theorem 4.2. THEOREM 4.4. Let G g w AP x 2, 2 be defined by Ž3.1. and Ž3.2. with h "g AP " and f 1 such that f 1 Ž j . s fq Ž j . e igj fy Ž j . ,

507

THE CORONA THEOREM

where either Ži.

"1 y y1 q fqg APq W , fy g APW , el fy g APW , and i z maxy g , 04 < inf max  < fq Ž z . e i z maxg , 04 < , < e i l z fy1 4 )0 y Ž z. e

zgP q

or Žii.

"1 q y1 y fyg APy W , fq g APW , ey l fq g APW , and i z miny g , 04 < inf max  < fy Ž z . e i z ming , 04 < , < eyi l z fy1 4 ) 0. q Ž z. e

zgP y

Then when the matrix-¨ alued function G satisfies condition Ži.: If g G 0, G has a canonical AP factorization with factors G " gi¨ en by Ž3.10. and Ž3.11., where fq is replaced by fq eg and f˜q, ˜ gq are corona y1 q solutions in H`q for the corona data fq eg , el fy g APW . If g - 0, G is Ž canonically. AP factorizable only simultaneously with eyi gj yf˜q Ž j . eyi lj fy Ž j .

0 e igj

,

y1 where f˜q, ˜ gqg APq W are corona solutions for the corona data fq, el y g fy g q APW . When the matrix-¨ alued function G satisfies condition Žii.: If g F 0, G has a canonical AP factorization with factors G " gi¨ en by Ž3.13. and Ž3.14., where fy is replaced by fy eg and f˜y, ˜ gy are corona y1 y solutions in H`y for the corona data fy eg , ey l fq g APW . If g ) 0, G is Ž canonically. AP factorizable only simultaneously with

e igj yf˜y Ž j . e i lj fq Ž j .

0 yigj

e

,

y1 where f˜y, ˜ gyg APy W are corona solutions for the corona data fy, ey l y g fq y g APW .

Proof. If g G 0 with condition Ži. or g F 0 with condition Žii. to obtain the assertion of the theorem it is enough to introduce, respectively, the y functions fq 1 s fq eg or f 1 s fy eg and to apply Theorem 4.2. Let g - 0 with condition Ži. Žanalogously if g ) 0 with condition Žii... i lj y1 Ž . q Ž . Let gq fy j . Since fqg APW , gq 1 s gq ey g , where gq j s e 1 g q q y1 "1 y APW , g 1 s ely g fy with l y g ) 0 and fy g APW , we obtain due to q Corollary 4.3 that for the corona data fq, gq 1 g APW there exist corona q solutions f˜q, ˜ gqg APW .

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BASTOS ET AL.

Then the matrix-valued function G is Žcanonically. AP factorizable only simultaneously with the matrix function G1 Ž j . s

s

0

fy1 y Žj .

yfy Ž j .

0

fq Ž j .

˜gq Ž j .

e i lj fq Ž j . e igj fy Ž j .

f˜q Ž j .

˜gq Ž j . =

s

ygq Ž j . eyi gj

fq Ž j .

ygq Ž j . eyi gj f˜q Ž j .

eyi gj yf˜q Ž j . rgq Ž j .

0 e igj

e

gq Ž j .

0 ygy1 q

0 yi lj

Žj.

fq Ž j . e igj

,

where yf˜q Ž j . rgq Ž j . s yf˜q Ž j . eyi lj fy Ž j . .

As a consequence of this theorem we may obtain the reduction procedure of w2x, which can be applied recursively Žsee w2x and w25x.. Let l ) 0, g - 0, and f Ž j . s fq Ž j . e igj , q where fqg APW and M Ž fq . / 0. The matrix-valued function

GŽ j . s

e i lj fq Ž j . e igj

0 eyi lj

satisfies all the conditions of Theorem 4.4Ži. with fys 1. Consequently, G is AP factorizable only simultaneously with the matrix function G1 Ž j . s

eyi gj yf˜q Ž j . eyi lj

0 e igj

and their partial AP indices coincide. Moreover, we can eliminate from yf˜q ey l the part with the spectrum outside xg , yg w. Thus the corona theorem is in the basis of the mentioned reduction procedure and on the other hand Theorem 4.4 with g - 0 and condition Ži. or with g ) 0 and

509

THE CORONA THEOREM

condition Žii. can be considered as a generalization of the reduction theorem from w2x. Theorem 4.2 gives conditions for canonical AP factorization G s Gq Gy, but the factors G " depend on the corona solutions that in general we do not know explicitly. However, before the end of this section we will obtain conditions under which the matrix dŽ G . s M Ž Gq . M Ž Gy . can be calculated independently of corona solutions. THEOREM 4.5.

Let G g w AP x 2, 2 be defined by Ž3.1. with f Ž j . s fq Ž j . fy Ž j . ,

where fq" 1 g APq,

y ey l fy1 q g AP ,

fy" 1 g APy,

y1 el fy g APq.

Then the matrix function G admits the canonical AP factorization G s Gq Gy , "1 "1 where Gq g w APqx 2, 2 , Gy g w APyx 2, 2 are defined by Ž3.16., Ž3.17., and

dŽ G. s

M Ž el fy1 y . M Ž fy .

y1 y1 y1 yM Ž fy1 q . M Ž fy . q M Ž el fy . M Ž ey l fq .

M Ž fq . M Ž fy .

M Ž fq . M Ž ey l fy1 q .

.

Proof. Directly from Corollary 3.3 because we know from Ž3.16. and Ž3.17. that by hypothesis the factors belong to wAP " x 2, 2 . THEOREM 4.6. representation

Let G g w AP x 2, 2 be defined by Ž3.1., where f has the

f Ž j . s fq Ž j . fy Ž j . y hq Ž j . e i lj y hy Ž j . eyi lj

Ž h "g AP " .

that satisfies condition Ži. of Theorem 4.2 and the representation y q i lj yi lj f Ž j . s fq y hy 0 Ž j . f 0 Ž j . y h0 Ž j . e 0 Žj .e

Ž h "0 g AP " .

that satisfies condition Žii. of Theorem 4.2. Let M Ž hy . s 0,

M Ž hq 0 . s 0,

and M Ž fq . y M Ž hq . M Ž gq . / 0

y y or M Ž fy 0 . y M Ž g 0 . M Ž h 0 . / 0,

Ž 4.6.

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BASTOS ET AL.

y Ž q.y1 . Then the matrix-¨ alued function G admits where gqs el fy1 y , g 0 s ey l f 0 a canonical AP factorization and dŽ G . is gi¨ en by y M Ž gq . M Ž fy . M Ž fq 0 . M Ž g0 . y 1

M Ž gq . M Ž fy .

M Ž fy . w M Ž fq . y M Ž hq . M Ž gq . x

M Ž fy . w M Ž fq . y M Ž hq . M Ž gq . x

MŽ

fq 0

. MŽ

gy 0

.

.

Proof. From Theorem 4.2, G has two canonical AP factorizations G s Gq Gy ,

y G s Gq 0 G0 ,

where G " are defined by Ž3.10. and Ž3.11., and G 0" have the form Ž3.13. and Ž3.14.. . Then, taking into account that M Ž hy . s 0 and M Ž hq 0 s 0, we get that dŽ G. s

1 yM Ž hq . =

s

0 1

M Ž f˜qq gq gq 2 . yM Ž ˜ gqy

fq gq 2

0

yM Ž fy1 y .

M Ž fy .

y1 yM Ž gy 2 . M Ž fy .

M Ž gq . M Ž fq .

.

M Ž gq . M Ž fy .

)

M Ž f . M Ž fq . y M Ž hq . M Ž gq .

)

y

and on the other hand, dŽ G. s

q yM Ž gq 20 . rM Ž f 0 .

y1rM Ž gq 20 .

M Ž fq 0 .

0

= = s

M Ž fy 0 .

M Ž gy 0 .

y y yM Ž ˜ gy 0 y g 20 f 0 .

y y M Ž f˜y 0 q g 20 g 0 .

1 yM Ž

hy 0

.

0 1

) y y M Ž fq M f y M Ž gy . Ž . 0 0 0 . M Ž h0 .

) , y M Ž fq . 0 M Ž g0 .

" ˜y q where g 2" are defined by Ž3.12. and g 20 s yry1 q P " rq f 0 f 0 el .

511

THE CORONA THEOREM

Then we have y y y M Ž fy . M Ž fq . y M Ž hq . M Ž gq . s M Ž fq 0 . M Ž f0 . y M Ž g0 . M Ž h0 .

. Ž . and from M Ž fy . / 0, M Ž fq 0 / 0 it follows that both numbers in 4.6 are nonzero. As a result dŽ G. s

)

M Ž gq . M Ž fy . M Ž fy . M Ž fq . y M Ž hq . M Ž gq .

MŽ

fq 0

. M Ž gy0 .

. Ž 4.7.

Since M Ž fq . y M Ž hq . M Ž gq . / 0 and det dŽ G . s 1, the unknown entry of Ž4.7. is y M Ž gq . M Ž fy . M Ž fq 0 . M Ž g0 . y 1

M Ž fy . M Ž fq . y M Ž hq . M Ž gq .

COROLLARY 4.7.

.

Let G be defined by Ž3.1. with f s fq fy

Ž 4.8.

such that f s f, q fqg APW ,

y fy" 1 g APW ,

inf max  < fq Ž z . < ,

zgP q

y1 q el fy g APW , y1 < e i l z fy

M Ž fq . / 0,

Ž z . < 4 ) 0.

Then G admits a canonical APW factorization and dŽ G. s

M Ž gq . M Ž fy .

< M Ž fy . M Ž gq . < 2 y 1 r M Ž fq . M Ž fy .

M Ž fq . M Ž fy .

M Ž fy . M Ž gq .

.

Proof. It suffices to apply Theorem 4.6 because from f s f the reprey q sentation Ž4.8. gives another representation f s fq 0 f 0 with f 0 s fy and y f 0 s fq . 5. CRITERIA FOR CANONICAL AP FACTORIZATION We shall see in this section that the sufficient conditions of Theorem 4.2 are also necessary after applying an appropriate transformation which we denote by w transform.

512

BASTOS ET AL.

For each w g C _  04 , we define the w transform as an algebraic isomorphism of the nonclosed subalgebra of G g wAPW x n, n with bounded spectrum, wAPTW x n, n , onto itself, which acts by the rule G ¬ Gw , where GŽ j . s

Ý Gn e inj ,

Ž 5.1.

n

Gw Ž j . s G Ž j y i ln w . s

Ý Gn w n e inj ,

Ž 5.2.

n

and ln w denotes an arbitrary value of logarithm. Let us begin by stating two auxiliary results. LEMMA 5.1. Let G g w AP x n, n , V Ž G " 1 . ; wyl, lx, 0 F l - `, and let G possess a canonical AP factorization G s Gq Gy . "1. "1. Then V Ž Gq ; w0, lx and V Ž Gy ; wyl, 0x. "1 Proof. Since G has a canonical AP factorization, by definition, Gq g "1 yx wAPqx n, n and Gy w g AP n, n . We have y1 Gqs GGy ,

y1 Gq s Gy Gy1 .

. ; w0, lx. Analogously we have Then V Ž Gq . ; w0, lx and V Ž Gy1 q y1 Gys Gq G,

y1 Gy s Gy1 Gq

. ; wyl, 0x. and V Ž Gy . ; wyl, 0x, V Ž Gy1 y LEMMA 5.2. Let G g w APW x n, n be in¨ ertible and V Ž G " 1 . ; wyl, lx, 0 F l - `. Then G possesses a canonical AP factorization if and only if for each w g C _  04 Ž equi¨ alently, for some w g C _  04., the matrix function Gw defined by Ž5.2. has a canonical AP factorization. In particular, dŽ G . s dŽ Gw .. Proof. Let G g wAPW x n, n possess a canonical AP factorization G s 1 w "x Gq Gy. Then by Theorem 2.5, G " " g APW n, n and from Lemma 5.1, "1. "1. Ž w x Ž w x V Gq ; 0, l and V Gy ; yl, 0 . Since the w transform, G ¬ Gw , is an algebraic isomorphism of wAPTW x n, n onto wAPTW x n, n , the canonical AP factorization G s Gq Gy is equivalent, for each w g C _  04 , to the canonical AP factorization Gw s Gwq Gwy . Obviously, then dŽ G . s dŽ Gw .. THEOREM 5.3.

Let G be defined by GŽ j . s

e i lj fŽ j .

0 , eyi lj

Ž 5.3.

513

THE CORONA THEOREM

where f g APW and V Ž f . ; wyl, lx, 0 - l - `. Then the following three conditions are equi¨ alent: Ži. Žii.

G has a canonical AP factorization; there is a complex constant w / 0 such that fw Ž j . s fq Ž j . fy Ž j . y hy Ž j . eyi lj ,

Ž 5.4.

where q , fqg APW

y1 q gqs el fy g APW ,

y fy" 1 g APW ,

y hyg APW ,

inf max  < fq Ž z . < , < gq Ž z . < 4 ) 0;

zgP q

Žiii.

there is a complex constant w / 0 such that fw Ž j . s fq Ž j . fy Ž j . y hq Ž j . e i lj ,

Ž 5.5.

where y fyg APW ,

y1 y gys ey l fq g APW ,

q fq" 1 g APW ,

q hqg APW ,

inf max  < fy Ž z . < , < gy Ž z . < 4 ) 0.

zgP y

Proof. Ži. « Žii.. By Theorem 2.5, a canonical AP factorization G s Gq Gy is automatically a canonical APW factorization. Thus we have y1 G s Gy , Gq

Ž 5.6.

qx "1 "1 w yx where Gq g wAPW 2, 2 , Gy g APW 2, 2 , and

y1 Gq s

gq 11 gq 21

gq 12 , gq 22

Gys

gy 11 gy 21

gy 12 gy 22

Ž 5.7.

. w x Ž y. w x Ž k, j s 1, 2. due to Lemma 5.1. with V Ž gq k j ; 0, l and V g k j ; yl, 0 Ž . Ž . From 5.6 and 5.7 , yi lj gq s gy 12 Ž j . e 12 Ž j . ,

Ž 5.8.

i lj y gq q gq 11 Ž j . e 12 Ž j . f Ž j . s g 11 Ž j . ,

Ž 5.9.

yi lj gq s gy 22 Ž j . e 22 Ž j . ,

Ž 5.10.

i lj y gq q gq 21 Ž j . e 22 Ž j . f Ž j . s g 21 Ž j . .

Ž 5.11.

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BASTOS ET AL.

Then q y yi lj gy , 12 Ž j . f Ž j . s yg 11 Ž j . q g 11 Ž j . e

Ž 5.12.

q y yi lj gy . 22 Ž j . f Ž j . s yg 21 Ž j . q g 21 Ž j . e

Ž 5.13.

. Ž y. Let us show that M Ž gy 12 / 0 or M g 22 / 0. We have y1 det Gq det G s det Gy ,

whence q q q y y y y gq 11 g 22 y g 12 g 21 s g 11 g 22 y g 12 g 21 .

Ž 5.14.

"1 q "1 y Ž Since det Gq g APW and det Gy g APW , 5.14. implies that q q q y y y y gq 11 g 22 y g 12 g 21 s g 11 g 22 y g 12 g 21 s C / 0.

Ž 5.15.

The last relation shows that the equalities q q y M Ž gy 12 . s M Ž g 12 ey l . s M Ž g 22 ey l . s M Ž g 22 . s 0

cannot hold simultaneously because then we have det Gys 0. . Let us assume that M Ž gy 12 / 0. Since the spectra of all the functions in Ž5.12. are bounded, we apply the w transform to Eq. Ž5.12. to obtain y yl yi lj ygq e s gy 11 , w Ž j . q g 11, w Ž j . w 12, w Ž j . fw Ž j . .

Let gy 12 Ž j . s

Ý an e inj .

nF0

. Since a0 s M Ž gy 12 / 0, let us write gy 12 , w Ž j . s

Ý an w n e inj s a0 Ž 1 q ay1 0 gw Ž j . . .

nF0

Fix « g Ž0, < a0 <. and choose n« - 0 such that

Ý

n«- n -0

< an < - «r2,

and w ) 1 such that

Ý

nF n«

< an < w n - w n«

Ý

n F n«

< an < - «r2.

515

THE CORONA THEOREM

We get < gw < -

Ý

n« - n -0

< an < w n q

Ý

nF n«

< an < w n - « - < a0 < .

y . " 1 g APW Then Ž gy and 12, w y fw Ž j . s ygq 11, w Ž j . Ž g 12, w Ž j . .

y1

y q gy 11, w Ž j . Ž g 12, w Ž j . .

y1

wy l eyi lj .

Ž 5.16. Let fqs gq 11, w ,

fy" 1 s y Ž gy 12, w .

y hys ygy 11, w Ž g 12, w .

y1

wy l ,

.1

,

gqs el fy1 y .

q y1 y We have fqg APW and hy, fy g APW . Applying the w transform to q y l yi lj y Ž5.8., we get that g 12, w Ž j . w e s g 12, w Ž j .. Then q yl gq Ž j . s ye i lj gy , 12, w Ž j . s yg 12 , w Ž j . w

Ž 5.17.

q whence gqg APW . Consequently, we can rewrite Ž5.16. in the form Ž5.4.. It remains only to show that

inf max  < fq Ž z . < , < gq Ž z . < 4 ) 0.

zgP q

Ž 5.18.

q Let us consider the corona problem for the corona data fq, gqg APW . Applying the w transform to Ž5.8. ] Ž5.11., from Ž5.17. and Ž5.15. we have

ygq Ž j . eyi lj s gy 12, w Ž j . ,

Ž 5.19.

yl yi lj Cy1 gq e s Cy1 gy 22 , w Ž j . w 22, w Ž j . ,

Ž 5.20.

fq Ž j . w le i lj y gq Ž j . w l fw Ž j . s gy 11, w Ž j . ,

Ž 5.21.

l i lj y1 y Cy1 gq q Cy1 gq g 21, w Ž j . , Ž 5.22. 21 , w Ž j . w e 22, w Ž j . fw Ž j . s C

From Ž5.15., y y y y1 1 s Cy1 Ž gy Ž gy11, w gy22, w y gy12, w gy21, w . . 11 g 22 y g 12 g 21 . s C

Then from Ž5.20., Ž5.21., Ž5.19., and Ž5.22. we have fq f˜qq gq ˜ gqs 1,

Ž 5.23.

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BASTOS ET AL.

where l ˜gqs Cy1 gq 21, w w .

f˜qs Cy1 gq 22, w ,

Since the corona problem Ž5.23. is solvable, we get Ž5.18.. Ži. « Žiii. The proof is analogous. Žii. « Ži. Since G " 1 g wAPW x 2, 2 and V Ž G " 1 . ; wyl, lx, from Lemma 5.2 it follows that G has a canonical AP factorization only simultaneously with

Gw Ž j . s

w le i lj

0 y l yi lj

fw Ž j .

w

e

w g C _  04 ,

,

or, equivalently, with e i lj fw Ž j .

˜w Ž j . s G

0 yi lj

e

.

Ž 5.24.

˜w defined by Ž5.24. with conditions Žii. has a From Theorem 4.2, G canonical AP factorization. Žiii. « Ži. The proof is analogous. By Theorem 4.2, under the conditions Žii. and Žiii. of Theorem 5.3, the ˜w s G˜wq G˜wy with matrix function Ž5.24. has a canonical AP factorization G " ˜w given by Ž3.10. and Ž3.11. in case Žii. and by Ž3.13. and Ž3.14. in factors G ˜wq . " 1 g wAPWq x2, 2 , case Žiii.. Actually Žsee the proof of Lemma 5.2., Ž G y " 1 y q " 1 y " ˜w . g wAPW x2, 2 and V ŽŽ G˜w . . ; w0, lx, V ŽŽ G˜w . 1 . ; wyl, 0x. Then ŽG according to Lemma 5.2 the matrix function Ž5.3. under the conditions Žii. and Žiii. has a canonical APW factorization G s Gq Gy with factors

˜wq Gqs diag  w l , 1 4 G

ž /

w y1

,

˜wy Gys G

ž /

wy 1

diag  1, wy l 4 . Ž 5.25.

Thus we have the following. COROLLARY 5.4. Under conditions Žii. and Žiii. of Theorem 5.3, the matrix function Ž5.3. has a canonical APW factorization G s Gq Gy with the

517

THE CORONA THEOREM

factors Gqs

Gys

w l Ž f˜qq gq gq 2 .

w l gq

yg˜qq fq gq 2

fq

, w y1

fy1 y hy

y1 y l yfy w

y1 fyq gy 2 fy hy

y1 y l ygy 2 fy w

Ž 5.26. w

y1

in case Žii., where f˜q, ˜ gq are corona solutions in H`q for the corona data fq, q " gqg APW and g 2 are gi¨ en by Ž3.12.; and Gqs

q yw l fy1 q g2

yw l fy1 q

q fqq hq fy1 q g2

y1 hq fq

fy

Gys

yg˜yq gy 2 fy

, w

y1

yl

gy w

Ž f˜yq gy2 gy . wy l

Ž 5.27. wy1

in case Žiii., where f˜y, ˜ gy are corona solutions in H`y for the corona data fy, y " gyg APW and g 2 are gi¨ en by Ž3.15.. As a consequence of the proof of Theorem 5.3, let us state a result about the solvability of the corona problem for functions of bounded spectrum Žcf. Corollary 4.3.. THEOREM 5.5.

Let

q APq W Ž l . s  f g APW : V Ž f . ; w 0, l x 4 ,

l G 0.

Ž . Ž . The corona problem for corona data fq, gqg APq W l with M gq ey l / 0 is qŽ . sol¨ able in APW l if and only if inf max  < fq Ž z . < , < gq Ž z . < 4 ) 0.

zgC

Ž 5.28.

Proof. We begin with the necessary condition. Let the corona problem qŽ . qŽ . be solvable in APW l for the corona data fq, gqg APW l . Then there q are functions f˜q, ˜ gqg APW Ž l. such that fq f˜qq gq ˜ gqs 1.

Ž 5.29.

Since the spectra of all functions of Ž5.29. are bounded, we may apply the w transform and, for each w g C _  04 , we have fwq f˜wq q gq gq w ˜ w s 1.

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BASTOS ET AL.

Thus, for each w g C _  04 , < inf max  < fwq Ž z . < , < gq w Ž z . 4 ) 0.

Ž 5.30.

zgP q

Moreover, fwq Ž z . s fq Ž z y i ln w . ,

gq w Ž z . s gq Ž z y i ln w . ,

and Ž5.30. can be rewritten as Ž5.28.. For the sufficient condition let condition Ž5.28. hold. qŽ . We want to show that, for the corona data fq, gqg APW l with the condition M Ž gq ey l . / 0, the corona problem fq f˜q gq 2 q gq ˜ 2 s1

Ž 5.31.

qŽ . is solvable in APW l. qŽ . y Since gqg APW l , gq ey l g APW . As M Ž gq ey l . / 0 it is possible to choose w ) 1 such that the function fwy s elrgq w becomes invertible in y APW . Consider the function

fw Ž j . s fwq Ž j . fwy Ž j . y hwy Ž j . eyi lj ,

Ž 5.32.

y Ž . w x where the function hy w g APW is chosen so that V fw ; yl, l . It is clear that q fwq g APW ,

Ž fwy .

"1

y g APW ,

y gq w s el Ž fw .

y1

q g APW ,

and from Ž5.28., < inf max  < fwq Ž z . < , < gq w Ž z. 4

zgP q

s infq max  < fq Ž z y i ln w . < , < gq Ž z y i ln w . < 4 ) 0. zgP

Then Theorem 4.2 with the consequent note implies that the matrixvalued function Ž5.3. with f s fw defined by Ž5.32. has a canonical AP "1 qx "1 w yx factorization, where the factors Gq g wAPW 2, 2 and Gy g APW 2, 2 are defined by

Gqs

q f˜wq q gq w g2

ygq w

˜ q

fwq gq 2

gq w fwq

,

Gys

0 fwy

y Ž fwy . ygy 2 Ž

y1

y1 fwy

.

1 yhy w

0 1 .

519

THE CORONA THEOREM

Here f˜wq , ˜ gq are corona solutions in H`q for the corona data fwq , w q qŽ . ˜q y gw g APW l and g 2" s yry1 q P " rq fw fw ey l . As we have seen after Theorem 4.2, the functions q f˜wq, 1 s f˜wq q gq w g2 ,

q q gq ˜gq w,1 s ˜ w y fw g 2

give a solution of the corona problem for fwq , gq w because q f˜wq, 1 fwq q ˜ gq w , 1 g w s 1. q Moreover, f˜wq, 1 , ˜ gq and, by Lemma 5.1, V Ž f˜wq, 1 . ; w0, lx, w , 1 g APW q VŽ ˜ gw , 1 . ; w0, lx. Ž ˜q . Ž gq . Then we may conclude that the functions f˜q gq 2 s fw , 1 w y1 , ˜ 2 s ˜ w , 1 wy 1 qŽ . belong to APW l and satisfy Ž5.31..

An analogous result is valid for y APW Ž l . s  f g APWy : V Ž f . ; w l , 0 x 4 ,

l F 0.

On the basis of Theorem 5.5 we get the following supplement to Theorem 5.3 and to Corollary 5.4. THEOREM 5.6. Let G be defined by Ž5.3. with f g APW and V Ž f . ; wyl, lx, 0 - l - `. Then the following assertions are equi¨ alent: Ži. Žii.

G has a canonical AP factorization G s Gq Gy; there is a complex constant w / 0 such that fw Ž j . s fq Ž j . fy Ž j . y hy Ž j . eyi lj ,

Ž 5.33.

where q fqg APW Ž l. ,

y1 q gqs el fy g APW Ž l. ,

y fy" 1 g APW ,

y hyg APW ,

inf max  < fq Ž z . < , < gq Ž z . < 4 ) 0;

zgC

Žiii.

there is a complex constant w / 0 such that fw Ž j . s fq Ž j . fy Ž j . y hq Ž j . e i lj ,

where y fyg APW Ž yl . ,

hqg APq W,

y1 y gys ey l fq g APW Ž yl . ,

q fq" 1 g APW ,

inf max  < fy Ž z . < , < gy Ž z . < 4 ) 0.

zgC

520

BASTOS ET AL.

Moreo¨ er, the factors G " are gi¨ en by Gqs

w l f˜q yg˜q

w l gq fq

,

Gys

w y1

fy1 y hy

yl yfy1 yw

fyy f˜q ey l hy

f˜q ey l wy l

w y1

Ž 5.34. Ž . in case Žii., where f˜q, ˜ gqg APq W l are corona solutions for the corona data qŽ . fq, gqg APW l , and Gqs

w l f˜y el

yw l fy1 q

fqy hq f˜y el

hq fy1 q

Gys

, w y1

fy

gy wyl

yg˜y

f˜y wy l

Ž 5.35. w y1

Ž . in case Žiii., where f˜y, ˜ gyg APy W yl are corona solutions for the corona yŽ . data fy, gyg APW yl . Proof. Ži. « Žii. Observing from the proof of Theorem 5.3 that the yl q corona data fqs gq g 12, w as well as the corona solu11, w and gqs yw y1 q y1 q qŽ . tions f˜qs C g 22, w and ˜ gqs C g 21, w w l belong to APW l , and using Theorem 5.5 instead of the usual version of the corona theorem, we get implication Ži. « Žii. by analogy with Theorem 5.3. Ži. « Žiii. is proved in the same manner. Žii. « Ži. Applying Theorem 5.5 we obtain that for the corona data fq, qŽ . qŽ . gqg APW l there exist corona solutions f˜q, ˜ gqg APW l . Then g 2 s y yŽ eyl f˜q . fyg APW and hence 1 G 2" 1 s "g 2

0 y 1 g APW

2, 2 .

Further, following the proof of Theorem 3.1 we get a canonical APW ˜w s G˜wq G˜wy , where factorization of the matrix function Ž5.24., G

˜wq s G

f˜q yg˜q

gq fq

,

˜wy s G

0

yfy1 y

fy

y1 yg 2 fy

1 yhy

0 1

˜ Žcf. Ž3.10. and Ž3.11... It remains to observe that yg 2 fy1 y s fq ey l and to apply Ž5.25.. Similarly, we get that Žiii. « Ži.. Note that the form of factors G " in Ž5.34. and Ž5.35. is simpler than in Ž5.26. and in Ž5.27.. After this paper was completed we learned from w26x that Corollary 4.3 admits the following strengthening.

521

THE CORONA THEOREM

q THEOREM 5.7. For corona data fq, gqg APW , the corona problem is q sol¨ able in APW if and only if Ž4.5. holds.

Then in Theorem 4.2 and in Corollary 5.4 we may take corona solutions f˜", ˜ g "g APW" and simplify formulas for g 2" , g 2" s

½

P " f˜q fy ey l yP

instead of Ž 3.12. ,

P " f˜y fq el yP

instead of Ž 3.15. ,

where P " are complementary projections on APW , Pq

ž Ý ˆŽ l. / f

l

el s

Ý fˆŽ l. el ,

l G0

Py

ž Ý ˆŽ l. / f

l

el s

Ý fˆŽ l. el .

l -0

REFERENCES 1. M. A. Bastos, Yu. I. Karlovich, A. F. dos Santos, and P. M. Tishin, The corona theorem and the canonical factorization of triangular AP-matrix functions}Effective criteria and explicit formulas, J. Math. Anal. Appl. 223 Ž1998., 523]550. 2. M. A. Bastos, Yu. I. Karlovich, I. M. Spitkovsky, and P. M. Tishin, On a new algorithm for almost periodic factorization, in ‘‘Operator Theory: Advances and Applications,’’ Vol. 103, pp. 53]74, Birkhauser, Basel, 1998. ¨ 3. M. A. Bastos and A. F. dos Santos, Convolution equations of the first kind on a finite interval in Sobolev spaces, Integral Equations Operator Theory 13 Ž1990., 638]659. 4. M. A. Bastos, A. F. dos Santos, and R. Duduchava, Finite interval convolution operators on the Bessel potential spaces H ps , Math. Nachr. 173 Ž1995., 49]63. 5. A. Bottcher and Yu. I. Karlovich, ‘‘Carleson Curves, Muckenhoupt Weights, and Toeplitz ¨ Operators,’’ Birkhauser, Basel, 1997. ¨ 6. A. Bottcher, Yu. I. Karlovich, and I. M. Spitkovsky, Toeplitz operators with semi-almost ¨ periodic symbols on spaces with Muckenhoupt weight, Integral Equations Operator Theory 18 Ž1994., 261]276. 7. L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 Ž1962., 547]559. 8. J. B. Garnett, ‘‘Bounded Analytic Functions,’’ Academic Press, New York, 1981. 9. M. Gelfand and I. M. Spitkovsky, Almost periodic factorization: Applicability of the division algorithm, forthcoming. 10. I. Gohberg and N. Krupnik, ‘‘One-Dimensional Linear Singular Integral Equations,’’ Vols. I and II, Birkhauser, Basel, 1992 ŽRussian original: Shtiintsa, Kishinev, 1973.. ¨ 11. R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 Ž1973., 227]251. 12. Yu. I. Karlovich, On the Haseman problem, Demonstratio Math. 26 Ž1993., 581]595. 13. Yu. I. Karlovich, The continuous invertibility of functional operators in Banach spaces, Dissertationes Math. 340 Ž1995., 115]136. 14. Yu. I. Karlovich and G. S. Litvinchuk, On some classes of semi-Fredholm operators, Iz¨ . Vyssh. Uchebn. Za¨ ed. 2, Ž1990., 3]16 Žin Russian.. 15. Yu. I. Karlovich and I. M. Spitkovsky, On the Noether property for certain singular integral operators with matrix coefficients of class SAP and the systems of convolution

522

16.

17.

18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

BASTOS ET AL.

equations on a finite interval connected with them, So¨ iet Math. Dokl. 27 Ž1983., 358]363. Yu. I. Karlovich and I. M. Spitkovsky, On the theory of systems of equations of convolution type with semi-almost periodic symbols in spaces of Bessel potentials, So¨ iet Math. Dokl. 33 Ž1986., 180]184. Yu. I. Karlovich and I. M. Spitkovsky, Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type, Math. USSR-Iz¨ . 34 Ž1990., 281]316. Yu. I. Karlovich and I. M. Spitkovsky, ŽSemi.-Fredholmness of convolution operators on the spaces of Bessel potentials, in ‘‘Operator Theory: Advances and Applications,’’ Vol. 71, pp. 122]152, Birkhauser, Basel, 1994. ¨ Yu. I. Karlovich and I. M. Spitkovsky, Factorization of almost periodic matrix functions, J. Math. Anal. Appl. 193 Ž1995., 209]232. Yu. I. Karlovich and I. M. Spitkovsky, Almost periodic factorization: An analogue of Chebotarev’s algorithm, Contemp. Math. 189 Ž1995., 327]352. B. Ja. Levin, ‘‘Distribution of Zeros of Entire Functions,’’ Am. Math. Soc., Providence, RI, 1964 ŽEnglish transl.. G. S. Litvinchuk and I. M. Spitkovsky, ‘‘Factorization of Measurable Matrix Functions,’’ Akademie-Verlag, Berlin, and Birkhauser, Basel, 1987. ¨ S. G. Mikhlin and S. Prossdorf, ‘‘Singular Integral Operators,’’ Springer-Verlag, Berlin, ¨ 1986. V. Yu. Novokshenov, Convolution equations on a finite segment and factorization of elliptic matrices, Math. Notes 27 Ž1980., 449]455. D. Quint, L. Rodman, and I. M. Spitkovsky, New cases of almost periodic factorization of triangular matrix functions, Michigan Math. J., to appear. L. Rodman and I. M. Spitkovsky, Almost periodic factorization and corona theorem, Indiana Uni¨ ersity Math. J., to appear. I. B. Simonenko, Some general questions of the theory of the Riemann boundary value problem, Math. USSR-Iz¨ . 2 Ž1968., 1091]1099. I. B. Simonenko, On the factorization and local factorization of measurable functions, So¨ iet Math. Dokl. 21 Ž1980., 271]274. I. M. Spitkovsky, Factorization of almost-periodic matrix-valued functions, Math. Notes 45 Ž1989., 482]488. J. Xia, Conditional expectations and the corona problem of ergodic Hardy spaces, J. Funct. Anal. 64 Ž1985., 251]274.