The Corona Theorem and the Canonical Factorization of Triangular AP Matrix Functions—Effective Criteria and Explicit Formulas

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

223, 523]550 Ž1998.

AY985969

The Corona Theorem and the Canonical Factorization of Triangular AP Matrix Functions}Effective Criteria and Explicit Formulas* 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 results of the companion paper, the authors obtain effective criteria for the existence of canonical factorization for new classes of triangular almost ilj periodic matrix functions of the form feŽ j . eyi0 l j . The important role in the proof of necessity in these criteria is played by a result on the stability of canonical AP factorizability obtained in the present paper. By finding corona solutions for a pair of binomial almost periodic functions, it is possible to calculate explicitly the factors of a canonical factorization for a class of considered matrix functions with trinomial f. Q 1998 Academic Press Key Words: canonical factorization; corona theorem; almost periodic matrix function.

1. INTRODUCTION This paper is a continuation of and companion to our previous paper w1x. To avoid repetition we will use definitions, notations, and results of w1x without additional comments. * Work sponsored by F. C. T. ŽPortugal. under grants Praxis XXIr2r2.1rMATr441r94 and Praxis XXIrBCCr4355r94. 523 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

524

M. A. BASTOS ET AL.

In this paper we derive effective criteria and explicit formulas for canonical factorization of triangular almost periodic ŽAP. matrix-valued functions. Having obtained, in the companion paper w1x, a criterion for the existence of canonical factorization for APW matrix-valued functions of the form GŽ j . s

e i lj fŽ j .

0 eyi lj

Ž 1.1.

and formulas for factorization factors in terms of corona solutions, we turn now to the question of calculating the factors and getting canonical factorization criteria for concrete classes of triangular APW matrix-valued functions in terms of their entries. The factorization problem for different classes of triangular APW matrix functions was studied in w10, 12, 13, 9, 2, 7, 16x. For applications of these results to the invertibility study of difference operators on a semi-axis or on a finite interval and to Fredholm theory of singular integral operators with semi-almost periodic matrix coefficients and of convolution type operators with semi-almost periodic matrix symbols, see, e.g., w10, 8, 11, 3, 9x. The paper is organized as follows. In Section 2 we state a result to be used later, about the stability of canonical APW factorizability of APW matrix-valued functions under small perturbations of their coefficients and exponents. In Section 3, for the periodic and APW versions of the corona theorem, some resutls on the determination of the corona solutions are presented. In Sections 4]6 the canonical AP factorization for concrete classes of APW matrix-valued functions is studied. A criterion for canonical AP factorization is given in Section 4 for a class of periodic functions and in Section 5 for classes of APW matrix-valued functions defined by Ž1.1. with f a quotient of polynomials. Finally, in Section 6, by applying the corona theorem we prove in an essentially simpler way a criterion for AP factorization for the matrix function Ž1.1. in the case of a trinomial f that was already established in w12x. Moreover, by using corona solutions obtained in Section 3 we calculate explicitly the factors of this factorization. We note that the factors are obtained here for the first time.

2. STABILITY OF APW FACTORIZABILITY Let APW be the algebra of absolutely convergent Fourier series Ý fˆŽ l. e i lj , where l runs through the spectrum V Ž f . ; R of f, and APW" s  f g APW : fˆŽ l . s 0 for "l - 0 4 .

CORONA THEOREM AND FACTORIZATION

525

We will use notations w X x n, n for the class of n = n matrix functions with entries in X, and 5 G 5 W s Ý l 5 Gl 5 if G s Ý Gl el g wAPW x n, n , where Gl are constant complex matrices. Below we will use the following result on the stability of the canonical APW factorizability of APW matrix-valued functions under small perturbations of coefficients and exponents Žcf. w18, Theorem 2x and w9, Theorem 5.6x; see also w13, Lemma 3.1 and Corollary 3.1x.. Let a matrix-valued function G g wAPW x n, n be represented in the form GŽ j . s

Ý Gh e i h j ,

j g R,

h

where Gh are constant complex matrices and h g R runs through an Abelian group H s H Ž G . generated by an at most countable set GG of real numbers g j which are linearly independent over the ring Z. Let us associate to each g j g GG a real number g jX and consider the Abelian group H9 generated by the set GGX s g jX : g j g GG 4 . The map n : GG ª GGX , g j ¬ g jX , can be extended to a homomorphism n : H ª H9 that induces a Banach algebra homomorphism

cn : APW Ž H .

Ý

n, n

ª APW Ž H9 .

Gh e i h j ¬

hgH

n, n ,

Gh e i n Ž h. j ,

Ý hgH

where wAPW Ž H .x n, n s  G g wAPW x n, n : V Ž G . ; H 4 . THEOREM 2.1.

Let G be a matrix-¨ alued function in w APW x n, n , GŽ j . s

Gh e i h j ,

Ý hgV Ž G .

and let G ha¨ e a canonical APW factorization. Then for each h g V Ž G . and for each g j g GG , there is a neighborhood UŽ Gh . of Gh in C n= n and a neighborhood Ug j of g j in R such that, for e¨ ery GXh g UŽ Gh . and for e¨ ery g jX g Ug j , the matrix-¨ alued function FŽ j . s

Ý

GXh e i n Ž h. j

hgV Ž G .

has a canonical APW factorization. Proof. Let G g wAPW x n, n have a canonical APW factorization G s Gq Gy ,

Ž 2.1.

"1 qx "1 w yx Ž " 1 . s I, where I stands with Gq g wAPW n, n , Gy g APW n, n , and M Gq for the identity matrix.

526

M. A. BASTOS ET AL.

First, we show that 1 V ŽG" " . ; H.

Ž 2.2.

Let y1 s Bqq Cq , Gq

Gys Byq Cy ,

Ž 2.3.

where B ", C "g wAPW"x n, n , V Ž B " . ; H, V Ž C " . l H s B, and M Ž Bq . s I. Ž . Since V Ž G " 1 . ; H, we get from Gy1 q G s Gy and 2.3 that Bq G s By, whence by Ž2.1., y1 Bq Gqs By Gy .

Ž 2.4.

qx y1 yx Ž . Since Bq Gqg wAPW , By Gy g wAPW n, n , and M Bq Gq s I, the equality Ž2.4. yields y1 Bq Gqs By Gy s I.

Ž y1 . Ž . Then Gy1 q s Bq, Gys By, and, hence, V Gq , V Gy ; H. Analogously y1 we may conclude that V Ž Gq ., V Ž Gy . ; H. As we have shown, there is an APW factorization Ž2.1. satisfying Ž2.2.. Then there is a sequence of matrix almost periodic polynomials Pk" such that 5 lim 5 Pk" y Gy1 " Ws0

kª`

and V Ž Pk". ; H. Then we may choose k g N such that 5 I y Pkq GPky 5 W - 12 .

Ž 2.5.

Fix this k and choose neighborhoods UŽ Gh . of coefficients Gh , h g V Ž G ., such that Pkq G y

ž

GXh e i h j Pky

Ý hgV Ž G .

/

-

1 2

Ž 2.6.

W

for all GXh g UŽ Gh . and all h g V Ž G .. From Ž2.5. and Ž2.6. we have I y Pkq

Ý

GXh e i h j Pky

hgV Ž G .

- 1.

Ž 2.7.

W

Choose now neighborhoods Ug j of generators g j g GG of the group H such that, for all g jX g Ug j ,

cn Ž Pk" . g APW"

n, n

.

Ž 2.8.

527

CORONA THEOREM AND FACTORIZATION

This is possible since V Ž Pk". ; H l R " and the sets V Ž Pk". are finite. Moreover, since Ž Pk".y1 g wAPW"x n, n , we get that 0 g V ŽŽ Pk".y1 . and the sets V ŽŽ Pk".y1 . R  04 ; R " are separated from zero. Hence

Ž cn Ž Pk" . .

y1

s cn

" y1 k

žŽ P

.

/g

APW"

n, n

.

Ž 2.9.

Since cn is a homomorphism of wAPW Ž H .x n, n into wAPW Ž H9.x n, n , where H9 s n Ž H . and since 5 cn Ž G .5 W F 5 G 5 W , we get from Ž2.7. that I y cn Ž Pkq .

GXh e i n Ž h. jcn Ž Pky .

Ý hgV Ž G .

F I y Pkq

W

GnX e i h j Pky

Ý hgV Ž G .

- 1.

Ž 2.10.

W

By Theorem 1.1 in w12x, from Ž2.10. it follows that the matrix-valued function F1 s cn Ž Pkq .

Ý

GXh e i n Ž h. jcn Ž Pky .

hgV Ž G .

has a canonical APW factorization F1 s F1q F1y , where V Ž F1". , V ŽŽ F1".y1 . ; H9 because V Ž F1 . ; H9. Then, by virtue of Ž2.8. and Ž2.9., the matrixvalued function FŽ j . s

GXh e i n Ž h. j

Ý hgV Ž G .

has a canonical APW factorization F s Fq Fy with factors Fqs Ž cn Ž Pkq . .

y1

F1q ,

Fys F1y Ž cn Ž Pky . .

y1

.

3. CORONA SOLUTIONS Let H`q be the Hardy space of bounded analytic functions on Pqs  z: Im z ) 04 , and let f 1 , . . . , f n be functions in H`q . By the corona theorem Žsee, e.g., w6x., the corona problem n

Ý f j f˜j s 1 js1

has solutions f˜1 , . . . , f˜n in H`q if and only if inf

max

zgP q js1, . . . , n

f j Ž z . ) 0.

Ž 3.1.

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M. A. BASTOS ET AL.

Corona problem Ž3.1. in the class APq; H`q is studied in w19x Žsee also w17x for its APW version.. If functions f 1 , . . . , f n g APq are periodic polynomials with period 2p , then the substitution t s e i j leads to usual polynomials fˆj Ž t . s f j Ž arg t . ,

j s 1, . . . , n,

on the unit circle G s  z: < z < s 14 , for which we get a version of the corona theorem for H`Ž D ., where D is the unit disk D s  z g C: < z < - 14 . In this case, if the data are polynomials, we make use of the following theorem. THE BEZOUT THEOREM Žsee, e.g., w4x.. Let f, g be polynomials that do ˜ ˜g such that not ha¨ e common roots. Then there exist polynomials f, ff˜q gg˜ s 1.

Ž 3.2.

In this section we present, for this periodic version, some useful results about the determination of the corona solutions. Given polynomials m

f Ž t . s ¨ m Ł Ž t y c i . s ¨ 0 q ¨ 1 t q ??? q¨ m t m ,

Ž 3.3.

is1 n

g Ž t . s wn Ł Ž t y s j . s w 0 q w 1 t q ??? qwn t n ,

Ž 3.4.

js1

we look for polynomials f˜Ž t . s a0 q a1 t q ??? qa ny1 t ny 1 ,

Ž 3.5.

˜g Ž t . s b 0 q b1 t q ??? qbmy1 t my 1 ,

Ž 3.6.

satisfying the corona condition Ž3.1.. To this end we can solve the algebraic system

¡!

n #

¨m

~

mqn

¨ my 1

¨m

.. . .. .

¨ my 1

..

.. . .. .

..

¨0

. .

0

.. ..

. . ¨m

¨0

¨ my1

..

¢

0

"!

.

..

.

wn

wny1 .. . .. . .. . w0

.. . .. . ¨0

0

m #

0

"

wn wny1 .. . .. . .. . w0

.. ..

.

. wn wny1 .. . .. . .. .. . . w0

a ny1 a ny2 .. . .. . a0 bmy1 bmy2 .. . b0

s

0 .. . .. . .. . .. . .. . .. .

.

0 1

Ž 3.7.

CORONA THEOREM AND FACTORIZATION

529

The determinant of this system is called the resultant of polynomials f, g and is denoted by ResŽ f, g .. As it is known Žsee, e.g., w15x, Chap. 4, Proposition 8.3., m

n

Res Ž f , g . s ¨ m wn Ł

Ł Ž ci y sj . .

Ž 3.8.

is1 js1

Then we arrive at the following sharpening of the Bezout theorem. PROPOSITION 3.1. Let f, g be polynomials gi¨ en by Ž3.3. and Ž3.4. with ˜ ˜g of the form Ž3.5. and ¨ m wn / 0. The corona problem Ž3.2. has a solution f, Ž3.6. if and only if ci / sj

for all i s 1, . . . , m and all j s 1, . . . , n.

Ž 3.9.

˜ ˜g can be found from Ž3.7. If Ž3.9. holds, then the coefficients of polynomials f, by the Cramer ´ rule. We emphasize that Ž3.9. is a criterion for the solvability of the corona problem Ž3.2. in the class of polynomials of arbitrary degree. If we look for f˜, ˜ g in the class of bounded analytic functions on the unit disk D, condition Ž3.9. must be fulfilled only for the zeros of f, g which lie in D. Now, for polynomial corona data f, g of some particular form, we construct, explicitly, corona solutions which will be used later. Naturally, their expressions look simpler than in the general case Žcf. Proposition 3.1.. PROPOSITION 3.2.

For the corona data

f Ž t . s t m y c,

g Ž t . s t n y s,

t g G,

where c, s g C; m, n g N, m - n and m, n are relati¨ ely prime, the corona problem Ž3.2. is sol¨ able in the class of polynomials if and only if c n / s m . If the latter holds, then one pair of corona solutions is f˜Ž t . s y

˜g Ž t . s

1 cn y sm

my1

kj

Ý

Ý

c ny 1yk s j t k myjn ,

js0 ksk jy1 q1

1

my1

cn y sm

Ý

c ny 1yk j s j t Ž k jq1. myŽ jq1. n ,

js0

where ky1 s y1,

k j s E Ž Ž j q 1 . nrm . , k my 1 s n y 1,

and EŽ j . denotes the integer part of j g R.

j s 0, . . . , m y 2;

530

M. A. BASTOS ET AL.

Proof. For given polynomials f, g we get from Ž3.7. or Ž3.8. that Res Ž f , g . s Ž y1 .

m

n Ž mq1 . n

c q Ž y1 . s m s Ž y1 .

n Ž mq1 .

Ž cn y sm. .

Then by Proposition 3.1, for these polynomials f, g the corona problem Ž3.2. is solvable in the class of polynomials if and only if c n / s m . Taking into account that Ž k my 1 q 1. m s mn we get

Ž t m y c . f˜Ž t . s

y1 c ys n

my1 m

Ý Ž yc ny 1yk

jy 1

s j t Ž k jy 1q1. myjn q c ny1yk j s j t Ž k jq1. myjn . ,

js0

Ž t n y s . ˜g Ž t . s

my1

1 cn y sm

Ý Ž c ny 1yk s j t Ž k q1. myjn y c ny1yk s jq1 t Ž k q1. myŽ jq1. n . , j

j

j

j

js0

whence

Ž t m y c . f˜Ž t . q Ž t n y s . ˜g Ž t . s 1.

We generalize Proposition 3.2 to the class of APW functions with bounded spectra. Let EŽ j . be the integer part of j g R and let E˜Ž j . be the greatest integer number which is less than j . LEMMA 3.3.

Let fq Ž j . s e i mj y c,

gq Ž j . s e i nj y s,

where c, s g C, 0 - m F n . Then for the sol¨ ability of the corona problem Ž .  Ž . w x4 it is necfq f˜qq gq ˜ gqs 1 in the class APq W n s f g APW : V f ; 0, n essary and sufficient that cn / sm,

if nrm is rational with m, n g N relati¨ ely prime such that

nrm s mrn; < c < n / < s < m,

if nrm is irrational.

531

CORONA THEOREM AND FACTORIZATION

Ž . If these conditions hold, then one pair of corona solutions f˜q, ˜ gqg APq W n ˜ which possess the additional property V Ž fq . ; w0, n w, V Ž ˜ gq . ; w0, mw, has, respecti¨ ely, the form:

Ž i. f˜q Ž j . s y

˜gq Ž j . s

1

my1

kj

cn y sm

Ý

Ý

js0 ksk jy1 q1

my1

1

c ny k jy1 s jexp i Ž Ž k j q 1 . m y Ž j q 1 . n . j

ž

Ý

cn y sm

c ny ky1 s jexp Ž i Ž k m y jn . j . ,

js0

/

with k j s E˜ŽŽ j q 1.nrm ., if nrm s nrm is rational; kj

`

Ž ii . f˜q Ž j . s y Ý

cyk y1 s jexp Ž i Ž k m y jn . j . ,

Ý

js0 ksk jy1 q1

`

˜gq Ž j . s Ý cyk jy1 s jexp ž i Ž Ž k j q 1 . m y Ž j q 1 . n . j / js0

with k j s E˜ŽŽ j q 1.nrm . and 0 f V Ž ˜ gq ., if nrm is irrational and < c < n ) < s < m ;

Ž iii .

f˜q Ž j . s

t jq1

`

Ý Ý

cty1 syjexp Ž i Ž ytm q jn . j . ,

js1 t s t jq1 `

˜gq Ž j . s y Ý ct j syjexp Ž i Ž yt j m q Ž j y 1 . n . j . js1

with t j s EŽŽ j y 1.nrm . and 0 f V Ž f˜q ., if nrm is irrational and < c < n - < s < m . Proof. Since 0 - m F n , the corona data fq, gq as well as the desired qŽ . corona solutions f˜q, ˜ gq belong to APW n . Then automatically ˜ gqg qŽ . qŽ . APW m because fqg APW m . By Theorem 5.5 from w1x, the corona qŽ . problem fq f˜qq gq ˜ gqs 1 is solvable in APW n if and only if inf max  fq Ž z . , gq Ž z .

zgC

4 ) 0.

Ž 3.10.

Ž Ž .. and gy1 Ž Ž .. of the Obviously, Ž3.10. holds if the preimages fy1 q D« c q D« s discs D« Ž c . s  z: < z y c < - « 4 and D« Ž s . s  z: < z y s < - « 4 do not intersect for a sufficiently small « ) 0. Investigate the fulfillment of Ž3.10. by comparing the zeros of the functions fq Ž z . s e i m z y c,

gq Ž z . s e i n z y s.

532

M. A. BASTOS ET AL.

The zeros of fq and gq are given by z s Ž arg c q 2p k y i log < c < . rm ,

k g Z,

Ž 3.11.

z s Ž arg s q 2p r y i log < s < . rn ,

r g Z.

Ž 3.12.

and Ž Ž .. and gy1 Ž Ž .. are contained in the sets Then fy1 q D« c q D« s F« Ž c . s

D

kgZ

ž

1

m

Y« Ž c . q

2p k

m

/

and

G« Ž s . s

D

kgZ

ž

1

n

Y« Ž s . q

2p k

n

/

,

respectively, where Y« Ž a . s z :

½

min

w g w0, 2 p .

arg Ž a q « e i w . - Re z -

max arg Ž a q « e i w . ,

w g w0, 2 p .

ylog Ž < a < q « . - Im z - ylog Ž max  < a < y « , 0 4 . .

5

If the imaginary parts of the roots in Ž3.11. and Ž3.12. are different, i.e., if < c < 1r m / < s < 1r n or, equivalently, < c < n / < s < m, then F« Ž c . l G« Ž s . s B for a small « ) 0 and hence Ž3.10. holds. If < c < n s < s < m but the number nrm is rational, then the condition cn / sm,

Ž 3.13.

where n, m are co-prime numbers in N such that nrm s nrm, denotes that

Ž arg c q 2p k . rm / Ž arg s q 2p r . rn for all k, r g Z. Consequently, Ž3.13. implies that the number Žarg c .rm y Žarg s .rn does not lie on the grid  2p lrŽ n m.: l g Z4 whence inf k , rgZ

Ž arg c q 2p k . rm y Ž arg s q 2p r . rn ) 0

and again Ž3.10. holds. It is easily seen that Ž3.10. is violated in other cases. Let us show that the formulas Ži., Žii., and Žiii. are corona solutions. In case Ži. the formulas for f˜q, ˜ gq follow from Proposition 3.2 under the replacement t s expŽ i njrn.. In cases Žii. and Žiii. all the expressions for f˜q, ˜ gq are well defined. It is easily seen that the series in case Žii. Žrespectively in case Žiii.. converges due to the inequality < c < n ) < s < m Žresp. < c < n - < s < m ..

533

CORONA THEOREM AND FACTORIZATION

Let nrm be irrational and < c < n ) < s < m. Then fq Ž j . f˜q Ž j . s

`

Ý cyk

jy 1 y1

s jexp i Ž Ž k jy1 q 1 . m y jn . j

ž

js0

y

/

`

Ý cyk y1 s jexp ž i Ž Ž k j q 1. m y jn . j / , j

js0

gq Ž j . ˜ gq Ž j . s y

`

Ý cyk y1 s jq1 exp ž i Ž Ž k j q 1. m y Ž j q 1. n . j / j

js0

q

`

Ý cyk y1 exp ž i Ž Ž k j q 1 . m y jn . j / , j

js0

whence, because ky1 s E˜Ž0. s y1, fq Ž j . f˜q Ž j . q gq Ž j . ˜ gq Ž j . s cyk y1 y1 exp Ž i Ž ky1 q 1 . mj . s 1. Obviously, 0 f V Ž ˜ gq .. Žiii. is considered analogously. Notice also that in this case the functions f˜q, ˜ gq can be rewritten in the form kj

y1

f˜q Ž j . s Ý

Ý

cyk y1 s jexp Ž i Ž k m y jn . j . ,

jsy` ksk jy1 q1

˜gq Ž j . s y

y1

Ý jsy`

cyk jy1 s jexp i Ž Ž k j q 1 . m y Ž j q 1 . n . j

ž

/

in view of the relation E Ž Ž j y 1 . nrm . s yE˜Ž Ž yj q 1 . nrm . y 1, i.e., t j s ykyj y 1. Finally, V Ž f˜q . ; w0, n w, V Ž ˜ gq . ; w0, mw since k j m - Ž j q 1.n F Ž k j q 1. m and hence 0 F Ž k jy1 q 1 . m y jn F k j m y jn s k j m y Ž j q 1 . n q n - n , 0 F Ž k j q 1. m y Ž j q 1. n s k j m y Ž j q 1. n q m - m .

Notice that in view of Theorem 5.5 in w1x, Ž3.10. is fulfilled automatically in cases Ži. ] Žiii. of Lemma 3.3 due to the explicit solutions produced.

534

M. A. BASTOS ET AL.

4. PERIODIC MATRIX FUNCTIONS Let G be defined by tn GŽ t . s

0

n

fq Ž t .

Ł fyk Ž t .

< t < s 1,

,

tyn

Ž 4.1.

ks1

where m

fq Ž t . s

Ý

fy k Ž t. s

ai t i ,

is0

mqn

Ý

i ks0

ski k tyi k ,

Ž 4.2.

and n ) 0, m ) 0. The matrix-valued function G defined by Ž4.1. with t s e i j is a particular case of Ž1.1., and in this section we get a criteria for canonical AP factorization for this class of periodic matrix functions. We emphasize that the criteria stated in Theorems 5.3 and 5.6 of w1x can give only sufficient conditions for the canonical AP factorizability of the matrix function Ž4.1. if the representation of the nondiagonal entry f of Ž4.1. in the form f s fq fyy hy ey l

Ž 4.3.

is fixed. Notice that such representation is not unique. LEMMA 4.1.

Let G be defined by Ž4.1. and Ž4.2. with t s e i j and < sk < - 1,

k s 1, . . . , n.

Ž 4.4.

Then the following two assertions are equi¨ alent: Ži. Žii. n  sk 4ks1 .

G has a canonical AP factorization. The roots of the polynomial fq in Ž4.2. do not belong to the set

Proof. Assume that condition Žii. is satisfied. Let n

f Ž t . s fq Ž t .

Ł fyk Ž t .

Ž 4.5.

ks1

and n

fy Ž t . s

Ł Ž 1 y sk ty1 .

y1

.

Ž 4.6.

ks1

Then we get f Ž t . q hy Ž t . tyn s fq Ž t . fy Ž t . ,

Ž 4.7.

535

CORONA THEOREM AND FACTORIZATION

where hy Ž t .

ž

Ý Ł js1

`

jy1

n

s fq Ž t .

ks1

fy k Ž t.

/ž

s jmqn tym

Ý

i js0

s ji j tyi j

n

/ž Ł Ž ksjq1

1 y sk ty1 .

y1

/

.

Define gq Ž t . s t n fy1 y Ž t. s

n

Ł Ž t y sk . .

Ž 4.8.

ks1

We have q fqg APW ,

q gqg APW ,

y fy" 1 g APW ,

y hyg APW .

The polynomials fq and gq do not have common roots and by the Bezout theorem there exist polynomials f˜q, ˜ gq of t s e i j such that fq f˜qq gq ˜ gqs 1. Then, by Theorem 4.2 from w1x, the matrix-valued function G admits a canonical AP factorization with periodic factors G " Žsee formulas Ž3.10. and Ž3.11. in w1x.. Conversely, assume that Ži. is satisfied and let us show Žii.. Suppose the polynomials fq and gq have common roots and S is the set of common roots of fq and gq. Denote by n 0 the cardinality of S . Let pŽ t . s

Ł Ž t y s. .

Ž 4.9.

sg S

The polynomials fq 1 s fqrp,

gq 1 s gqrp

Ž 4.10.

do not have common roots, and by the Bezout theorem, there are polynoq˜ q q q mials f˜q gq g 1 s 1. 1 and ˜ 1 such that f 1 f 1 q g 1 ˜ We have from Ž4.8. and Ž4.10. that fq Ž t . fy Ž t . s

fq Ž t . gq Ž t .

tn s

fq 1 Ž t. gq 1 Ž t.

t n.

In view of Ž4.5., Ž4.7., and Ž4.11. the matrix function G1 Ž t . s

fq 1

tn Ž t . t nrgq 1 Ž t.

0 tyn

has a canonical AP factorization simultaneously with Ž4.1..

Ž 4.11.

536

M. A. BASTOS ET AL.

Let yn qn 0 gy Ž t . s gq s gq Ž t . tyn r Ž p Ž t . tyn 0 . . 1 Ž t. t

Since p is a polynomial of degree n 0 ) 0, we get from Ž4.8., Ž4.9., and Ž4.4. "1 that gy g APy. Then the matrix function G2 Ž t . s

fq 1 Ž t.

ygq 1 Ž t.

˜gq 1 Ž t.

f˜q 1 Ž t.

gy Ž t .

0

G1 Ž t .

ygy1 y

Ž t.

0

also has a canonical AP factorization. On the other hand, fq 1 Ž t. gq 1

˜ Ž t. s

ygq 1 Ž t. fq 1

˜ Ž t.

G1 Ž t .

gy Ž t .

0 ygy1 y

tyn 0 ˜ Ž t . tynrgy Ž t .

yfq 1

Ž t.

0

0 t n0

Ž 4.12.

and since n 0 ) 0, the matrix-valued function on the right of Ž4.12. has an AP factorization with AP indices "n 0 / 0. Thus we get a contradiction, and assertion Žii. is proved. THEOREM 4.2. Let G be a periodic matrix function defined by Ž4.1., with fq and fy defined by Ž4.2. and t s e i j . Then G has a canonical AP k factorization if and only if the roots of the polynomial fq do not belong to the n set  sk 4ks1 . Proof. Since the spectrum of the function f given by Ž4.5. is bounded and the w transform gives Gw Ž t . s GŽ w t ., it follows from Lemma 5.2 in w1x that the matrix functions GŽ t . s

ž

tn f Ž t.

0

yn

t

/

Gw Ž t . s

,

˜w Ž t . s G

ž

tn f Ž wt.

ž

w nt n f Ž wt.

0 tyn

0 w t

yn yn

/

,

/

admit canonical AP factorizations only simultaneously. Choose w such that ˜w . < sk wy1 < - 1 for all k s 1, . . . , n. It remains to apply Lemma 4.1 to G With the help of Proposition 3.1 from this paper and Corollary 5.4 or Theorem 5.6 taken from w1x, it is possible to construct explicit canonical AP factorizations for matrix functions of the form Ž4.1..

537

CORONA THEOREM AND FACTORIZATION

5. APW MATRIX FUNCTIONS The criteria obtained in w2x allow us to identify new classes of APW matrix functions that have a canonical APW factorization. The next theorem gives one of them. Again the mentioned criteria give only the sufficient conditions for the canonical AP factorizability of matrix functions considered below. To prove the necessity of these conditions we will use Theorem 2.1 and Lemma 4.1. Let G g w APW x 2, 2 be defined by Ž1.1. with

THEOREM 5.1.

n

f Ž j . s exp Ž igj .

m

Ł Ž ak exp Ž i mk j . y bk . Ł Ž c j exp Ž in j j . y d j . ,

ks1

js1

Ž 5.1. where g g R; a k bk / 0, k s 1, . . . , n; c j d j / 0, j s 1, . . . , m; < n 1 < q ??? q< nm < F l ,

Ž 5.2.

and either

g G 0;

m k ) 0,

k s 1, . . . , n;

n j - 0,

j s 1, . . . , m, Ž 5.3.

m k - 0,

k s 1, . . . , n;

n j ) 0,

j s 1, . . . , m. Ž 5.4.

or

g F 0;

Then the matrix-¨ alued function G has a canonical AP factorization if and only if the following conditions hold: Ži. < c j dy1 < - 1, j s 1, . . . , m; j Žii. g s 0 if < n 1 < q ??? q< nm < - l; Žiii. for each k s 1, . . . , n and each j s 1, . . . , m, bk

1r < m k <

exp

ak /

cj dj

ž

i Ž arg Ž bkra k . q 2pa . < mk <

1r < n j <

exp

ž

/

i Ž arg Ž c jrd j . q 2pb .
/

538

M. A. BASTOS ET AL.

for all a , b g N if m k , n j are both rational; bk

1r < m k <

/

ak

cj

1r < n j <

dj

otherwise. Proof. We consider only the case

g G 0,

m k ) 0,

k s 1, . . . , n,

n j - 0,

j s 1, . . . , m;

the other situation is studied analogously. Let the three conditions Ži., Žii., and Žiii. be satisfied. Consider fy Ž j . s Ž y1 .

m

m

d1 d 2 ??? d m Ł Ž 1 y c j dy1 j exp Ž i n j j . . . js1

y We have fy" 1 g APW from Ži.. Ž . Rewrite 5.1 in the form

f Ž j . s fq Ž j . e i ljrgq Ž j . , where n

fq Ž j . s a1 a2 ??? a n exp Ž igj .

Ł Ž exp Ž i mk j . y ay1 k bk . ,

ks1

gq Ž j . s exp Ž i lj . rfy Ž j . m

s Ž y1 . d1 d 2 ??? d m exp Ž i Ž l q n 1 q ??? qnm . j . m

= Ł Ž exp Ž i < n j < j . y c j dy1 j .. js1

Obviously, fqg Žiii. we get that

q APW ,

q gqg APW in view of Ž5.2.. Moreover, from Žii. and

inf max  fq Ž z . , gq Ž z .

zgP q

4 ) 0.

Ž 5.5.

Indeed, consider the following functions in H`q :

w0 Ž j . s

½

exp Ž igj . , 1

c0 Ž j . s

½

1,

if l q n 1 q ??? qnn s 0,

exp Ž i Ž l q n 1 q ??? qnn . j . ,

if l q n 1 q ??? qnn ) 0;

w k Ž j . s a k exp Ž i m k j . y bk

Ž k s 1, . . . , n . ,

c j Ž j . s c j y d j exp Ž yi n j j .

Ž j s 1, . . . , m . .

539

CORONA THEOREM AND FACTORIZATION

Then fqs w 0 w 1 ??? wn ,

gqs c 0 c 1 ??? cm ,

and, by virtue of Žii. and Žiii. and of the proof of Lemma 3.3, inf max  w k Ž z . , c j Ž z .

zgP q

4 )0

Ž 5.6.

for every k s 0, 1, . . . , n and every j s 0, 1, . . . , m. It remains to observe that inf max  w 0 Ž z . w 1 Ž z . ??? wn Ž z . , c 0 Ž z . c 1 Ž z . ??? cm Ž z .

zgP q

G infq min max w k Ž z .

½

k, j

zgP

nq1

, cj Ž z .

G min 1, min infq max w k Ž z .

½

½

k , j zgP

½

nq1

mq 1

5

, cj Ž z .

G min 1, min infq max  w k Ž z . , c j Ž z .

ž

k , j zgP

4

mq 1

55

max  m , n4q1

4/

5

,

which implies Ž5.5. in view of Ž5.6.. Then by Theorem 4.2 from w1x, the matrix-valued function G has a canonical AP factorization. Conversely, let the matrix-valued function G admit a canonical AP factorization. Since G g wAPW x 2, 2 , this factorization is automatically a canonical APW factorization w18x. Assume that at least one of the conditions Ži. or Žii. is violated. Then a small perturbation of coefficients in Ž5.1. which preserves Žsee, e.g., Theorem 2.1. the canonical APW factorizability of the perturbed matrix func˜ s ef˜l e 0 leads to the fulfillment of Žiii. for f.˜ So we may suppose tion G yl that Žiii. holds for the original function f. Moreover, we may suppose that < c j dy1 < / 1 for all j s 1, 2, . . . , m. j Thus let us assume that Žiii. holds and that

ž

/

< c j dy1 < j

½

- 1,

for j g J 0 ;  1, 2, . . . , m4 ,

) 1,

for j g J1 s  1, 2, . . . , m4 R J 0 .

Then y1

m

žŁŽ js1

c j exp Ž i n j j . y d j .

/

y s fq 0 Ž j . exp Ž i n 0 j . f 0 Ž j . ,

540

M. A. BASTOS ET AL.

where fq 0 Ž j . s1

Ł Ž c j y d j exp Ž yin j j . . ,

jgJ 1

fy 0 Ž j . s1

Ł Ž c j exp Ž in j j . y d j . ,

jgJ 0

and

n0 s y

nj .

Ý jgJ 1

q Ž y. " 1 y . " 1 g APW We have Ž fq , f0 g APW , and, if J1 / B, n 0 ) 0. Then 0 for f defined by Ž5.1., we get the representation i lj f Ž j . s fq rgq 1 Žj .e 1 Ž j .,

where q fq 1 Ž j . s fq Ž j . f 0 Ž j . exp Ž i n 0 j .

n

s exp i g y

žž

Ý

nj j

/

jgJ 1

/

?

Ł Ž ak exp Ž i mk j . y bk . _

ks1

Ł Ž c j y d j exp Žyin j j . .

y1

,

jgJ 1

y gq 1 Ž j . s exp Ž i lj . rf 0 Ž j .

s exp i l q

žž

Ý

nj j

jgJ 0

/

/ŁŽ jgJ 0

c j y d j exp Ž yi n j j . . ,

q q and fq 1 , g 1 g APW . Let

d s min g y

½

Ý jgJ 1

nj , l q

Ý jgJ 0

nj .

5

If Ži. or Žii. is violated we get that d ) 0. Assume d ) 0 and define q yi dj fq , 2 Ž j . s f1 Ž j . e q yi dj gq . 2 Ž j . s g1 Ž j . e q q q q Ž . Then fq 2 , g 2 g APW and f s f 2 el rg 2 . Taking into account iii and the " q relation ŽŁ j g J1 Ž c j y d j expŽyi n j j ... 1 g APW we have q inf max  fq 2 Ž z . , g2 Ž z .

zgP q

4 ) 0.

541

CORONA THEOREM AND FACTORIZATION

y . " 1 s Ž fy . . 1 g APW Since l y d G 0 and Ž eyl q d gq , Corollary 4.3 from 2 0 q q q ˜q w1x ensures that there exist functions f˜2 , ˜ g 2 g APW such that fq 2 f2 q q q g 2 g˜2 s 1. q If we have corona solutions f˜q gq 2 , ˜ 2 g APW , the canonical AP factorizability of G implies the canonical AP factorizability of G1 defined by

fq 2 Žj .

G1 Ž j . s

ygq 2 Žj .

gq 2

f˜q 2 Žj .

˜ Žj. =

fq 2

0

1rfy 0 Žj .

yfy 0 Žj .

0

eyi dj y yi lj yf˜q 2 Ž j . f0 Ž j . e

s

e i lj Ž j . e i ljrgq 2 Žj .

0 e i dj

0 eyi lj

.

y " " Since d ) 0 and f˜q 2 f 0 ey l g APW , there exist functions h 0 g APW such that y yi lj yi dj i dj f˜q s hq q hy . 2 Ž j . f0 Ž j . e 0 Žj .e 0 Žj .e

This implies the canonical AP factorizability of G2 Ž j . s s

hq 0

1

Žj.

eyi dj 0

eyi dj yi lj ˜ Ž j . fy 0 Žj .e

0 1

yfq 2

0

0 e i dj

1 hy 0 Žj .

0 1

,

e i dj

which is impossible. Now let Ži. and Žii. hold, but assume Žiii. is violated. Since G defined by Ž1.1. remains canonically APW factorizable under small perturbations of the coefficients and under small perturbations of the exponents due to Theorem 2.1, we can find rational exponents ˜, g˜ , m l ˜ k , n˜j , k s 1, . . . , n, j s 1, . . . , m and coefficients a˜k , ˜bk , ˜c k , d˜k , different from zero, in all sufficiently small neighborhoods of the initial exponents and coefficients such that all the conditions of Theorem 5.1 except Žiii. are satisfied and at least for one pair Ž k, j . and Ž a , b . g N = N,

˜bk a ˜k

1r < m ˜k<

˜c j s d˜j

1r < n˜ j <

,

ž

arg

˜bk a ˜k

q 2pa

/


ž

˜c j q 2pb d˜j

/

< n˜j < .

542

M. A. BASTOS ET AL.

˜ can be written in a periodic The perturbed matrix-valued function G ˜ does not admit a form Ž4.1. and we conclude from Lemma 4.1 that G canonical AP factorization which contradicts the stability of the canonical APW factorizability property under small perturbations of the coefficients and exponents. COROLLARY 5.2.

Let G be defined by Ž1.1. with n

f Ž j . s a0 exp Ž igj . =

Ł Žexp Ž i mk j . y bk .

ks1

Ý

c1t 1 ct2 2 ??? ctmm exp Ž i Ž n 1t 1 q ??? qnmtm . j . , Ž 5.7.

Žt 1 , . . . , t m .g R

where g g R, a0 / 0, b1 b 2 ??? bn / 0, c1 c 2 ??? c m / 0, < n 1 < q ??? q< nm < F l, R s  Ž t 1 , . . . , tm . : < n 1
t j g  0, 1, . . . 4 , j s 1, . . . , m4 , and either condition Ž5.3. or Ž5.4. of Theorem 5.1 is satisfied. Then G has a canonical AP factorization if and only if Ži. g s 0 in case < n 1 < q ??? q< nm < - l, Žii. for each k s 1, . . . , n and each j s 1, . . . , m, < bk < 1r < m k < exp Ž i Ž arg bk q 2pa . r< m k < . / < c j < 1r < n j < exp Ž i Ž arg c j q 2pb . r< n j < .

for all a , b g N

if m k , n j are both rational; < bk < 1r < m k < / < c j < 1r < n j < , otherwise. Proof. Consider G satisfying condition Ž5.3.. Since the spectrum V Ž f . of the function f given by Ž5.7. is bounded, it follows from Lemma 5.2 in w1x that the matrix-valued function G has a canonical AP factorization if and only if the matrix function

˜w Ž j . s G

e i lj fw Ž j .

0 yi lj

e

,

w g C R  04 ,

543

CORONA THEOREM AND FACTORIZATION

with fw Ž j . s a0 w g exp Ž igj . =

n

Ł Ž w m exp Ž i mk j . y bk . k

ks1

c1t 1 ct2 2 ??? ctmm Ž w exp Ž i j . .

Ý

n 1t 1 q ??? q n mt m

Žt 1 , . . . , t m .g R

has a canonical AP factorization for every w g C R  04 . Then we may choose w ) 0 such that < c j w n j < - 1 for all j s 1, 2, . . . , m, y and find hyg APW such that

fw , h Ž j . s fw Ž j . q hy Ž j . exp Ž yi lj . m

s a0 Ž y1 . w g exp Ž igj . n

= Ł Ž w m k exp Ž i m k j . y bk . ks1

m

Ł Ž c j w n exp Ž in j j . y 1 . . j

js1

˜w and G˜w , h defined by Ž1.1. with f Ž j . s Since the matrix functions G fw , hŽ j ., have canonical AP factorizations only simultaneously, we get from Theorem 5.1 the assertion of the corollary. 6. TRINOMIAL MATRIX FUNCTIONS In this section we obtain a criterion of canonical AP factorizability and explicit expressions of the AP factors for the matrix-valued function GŽ j . s

e i lj cy1 eyi nj y c 0 q c1 e i a j

0 yi lj

e

,

Ž 6.1.

where cy1 c 0 c1 / 0,

0 - a,n - l s a q n.

Ž 6.2.

A criterion of canonical AP factorization for Ž6.1. is already known Žsee w12, Theorem 5.1x., but here we prove it in an essentially simpler way. We emphasize that if G is defined by Ž6.1. and does not have canonical AP factorization, it is not AP factorizable at all Žsee w12x..

544

M. A. BASTOS ET AL.

According to Lemma 3.3 define k j s E˜Ž Ž j q 1 . lra . ,

t j s E Ž Ž j y 1 . lra . ,

j g Z. Ž 6.3.

Let also c " 1, 0 s c " 1 cy1 0 , THEOREM 6.1.

s s cy1 cy1 1 .

Let G be defined by Ž6.1. and Ž6.2.. Then the matrix-

¨ alued function G has a canonical AP factorization if and only if the following

inequalities hold: m c1k cy1 / c 0kq m

Ž 6.4.

if b s nra is rational, where k, m g N are relati¨ ely prime numbers such that nra s krm; < c1 < b < cy1 < / < c 0 < bq1

Ž 6.5.

if b s nra is irrational. The factors of a canonical AP factorization G s Gq Gy ha¨ e the following form: Ži. If b is rational and Ž6.4. holds, then y Gq Ž j . s y

cy1 0 m cyn 1, 0 y s

1 m cyn 1, 0 y s

my1

kj

Ý

Ý

js0 ksk jy1 q1 my1

Ý

cynqk s jexp Ž i Ž k a y j l . j . 1, 0

qk jq1 j cyn s exp i Ž Ž k j q 1 . a y Ž j q 1 . l . j 1, 0

ž

js0

/

exp Ž i lj . y s , c1exp Ž i aj . y c 0 yc0 s q cy1 s exp Ž yi nj . Gy Ž j . s

1y

kj ynqk jq1 iŽ k a yŽ jq1. l . j Ý my1 s e js0 Ý ksk jy 1 q2 c1, 0 m cyn 1, 0 y s

s exp Ž yi lj . y 1 ycy1 0

1 kj ynqk j iŽ k a yŽ jq1. l . j Ý my s e js0 Ý ksk jy 1 q1 c1, 0

cyn 1, 0

ys

m

where n, m g N are co-prime numbers such that lra s nrm.

,

545

CORONA THEOREM AND FACTORIZATION

Žii.

If b is irrational and < c1 < b < cy1 < - < c 0 < bq1 , then ycy1 0

Gq Ž j . s

`

kj

Ý

Ý

js0 ksk jy1 q1

c1,k 0 s jexp Ž i Ž k a y j l . j .

`

j Ý c1k, q1 0 s exp ž i Ž Ž k j q 1 . a y Ž j q 1 . l . j /

y

j

js0

exp Ž i lj . y s ,

Ž 6.6.

.

Ž 6.7.

c1exp Ž i aj . y c 0 yc0 s q cy1 s exp Ž yi nj . Gy Ž j . s

1y

`

kj

Ý

Ý

c1,k 0 s jq1 exp Ž i Ž k a y Ž j q 1 . l . j .

js0 ksk jy1 q2

s exp Ž yi lj . y 1 ycy1 0 Žiii.

`

kj

Ý

Ý

js0 ksk jy1 q1

c1,k 0 s jexp Ž i Ž k a y Ž j q 1 . l . j .

If b is irrational and < c1 < b < cy1 < ) < c 0 < bq1, then cy1 0

Gq Ž j . s

`

Ý

t jq1

`

Ý Ý

js1 t s t jq1

t yj cy 1, 0 s exp Ž i Ž yta q j l . j . exp Ž i lj . y s

t j yj cy 1, 0 s exp

,

Ž i Žyt j a q Ž j y 1. l . j .

c1exp Ž i aj . y c 0

js1

Ž 6.8. yc0 s q cy1 s exp Ž yi nj . Gy Ž j . s

1q

t jq1 y1

`

Ý Ý

js1 t s t jq1

t yjq1 cy exp Ž i Ž yta q Ž j y 1 . l . j . 1, 0 s

s exp Ž yi lj . y 1 cy1 0

`

t jq1

Ý Ý

js1 t s t jq1

t yj cy 1 , 0 s exp Ž i Ž yta q Ž j y 1 . l . j .

.

Ž 6.9.

546

M. A. BASTOS ET AL.

Proof. From Lemma 5.2 in w1x it follows that the matrix-valued function G admits a canonical AP factorization if and only if the matrix function

˜w Ž j . s G

exp Ž i lj . cy1 wy n exp Ž yi nj . y c 0 q c1 w a exp Ž i aj .

0 , exp Ž yi lj .

where w ) 0, has a canonical AP factorization. Let us choose w ) 0 such that < cy1 cy1 < - w aq n 1 and define

˜cy1 s cy1 wyn ,

˜c1 s c1 w a .

˜w coincides with Then G ˜Ž j . s G

e i lj ˜cy1 eyi nj y c0 q ˜c1 e i a j

0 , eyi lj

˜ has a canonical AP < - 1, and it remains to prove that G where < ˜ cy1 ˜ cy1 1 factorization. Obviously, `

˜cy1 eyi nj y c0 q ˜c1 e i a j s Ž ˜c1 e i a j y c0 . Ý s k eyi k lj y hy Ž j . eyi lj ,

ž

/

ks0

< < where s s ˜ cy1 ˜ cy1 1 , s - 1, and `

hy Ž j . s Ž yc0 s q ˜ c1 s 2 eyi nj .

Ý s k eyi k lj . ks0

Then the function f˜Ž j . s ˜ cy1 eyi nj y c 0 q ˜ c1 e i a j is represented in the form f˜s fq fyy hy ey l , where fq Ž j . s ˜ c1 e i a j y c 0 s ˜ c1 Ž e i a j y ˜ cy1 1 c0 . , fy Ž j . s Ž 1 y seyi lj .

y1

,

y1 gq Ž j . s e i lj fy Ž j . s e i lj y s.

547

CORONA THEOREM AND FACTORIZATION

y qŽ . y qŽ . We have hyg APW , fqg APW l , fy" 1 g APW , and gqg APW l . On the other hand,

inf max  fq Ž z . , gq Ž z .

zgC

4 )0

when

Ž ˜cy1 1 c0 .

n

/ sm

Ž 6.10.

if lra is rational and n, m are co-prime numbers in N such that lra s nrm, and <˜ < l / < s< a cy1 1 c0

Ž 6.11.

if lra is irrational Žsee the proof of Lemma 3.3.. Note that Ž6.4. and Ž6.5. are equivalent to Ž6.10. and Ž6.11. because n y m s k and lra s b q 1. Then if Ž6.4. and Ž6.5. are satisfied, the ˜ has a canonical AP factorization in view of matrix-valued function G w x Theorem 5.6 in 1 . Now let us calculate the factors when G has a canonical AP factorization. First suppose that w s 1, i.e., s s cy1 cy1 and 1 ,

˜c " 1 s c " 1 ,

< s < - 1.

Since G satisfies the conditions Žii. of Theorem 5.6 w1x, the factors G " are given by Gq Ž j . s

Gy Ž j . s

f˜q Ž j .

gq Ž j .

yg˜q Ž j .

fq Ž j .

,

Ž 6.12.

hy Ž j . rfy Ž j .

y1rfy Ž j .

yf˜q Ž j . eyi lj hy Ž j . q fy Ž j .

f˜q Ž j . eyi lj

, Ž 6.13.

qŽ . where f˜q, ˜ gqg APW l are corona solutions for the corona data fq, gqg qŽ . APW l . Let k j , t j be given by Ž6.3. and c " 1, 0 s cy1 0 c " 1. qŽ . From Lemma 3.3 we get the following corona solutions f˜q, ˜ gqg APW l. If lra s nrm, then

f˜q Ž j . s y

˜gq Ž j . s

cy1 0 m cyn 1, 0 y s

my1

kj

Ý

Ý

js0 ksk jy1 q1

1

my1

m cyn 1, 0 y s

Ý js0

cynqk s jexp Ž i Ž k a y j l . j . , 1, 0

qk jq1 j cyn s exp i Ž Ž k j q 1 . a y Ž j q 1 . l . j . 1, 0

ž

/

548

M. A. BASTOS ET AL.

If lra is irrational and < c1 < b < cy1 < - < c0 < bq1 , then f˜q Ž j . s ycy1 0

`

kj

Ý

Ý

js0 ksk jy1 q1

c1,k 0 s jexp Ž i Ž k a y j l . j . ,

`

j ˜gq Ž j . s Ý c1,k jq1 0 s exp ž i Ž Ž k j q 1 . a y Ž j q 1 . l . j / .

js0

If lra is irrational and < c1 < b < cy1 < ) < c0 < bq1 , then f˜q Ž j . s cy1 0

t jq1

`

Ý Ý

js1 t s t jq1

t yj cy 1, 0 s exp Ž i Ž yta q j l . j . ,

`

t j yj ˜gq Ž j . s y Ý cy 1, 0 s exp Ž i Ž yt j a q Ž j y 1 . l . j . .

js1

This gives Gq by Ž6.12.. It can be checked directly that yf˜q Ž j . eyi lj hy Ž j . q fy Ž j . s Ž 1 y f˜q Ž j . eyi lj hy Ž j . rfy Ž j . . fy Ž j .

¡

1y

1

my1

kj

m cyn 1, 0 y s

Ý

Ý

js0 ksk jy1 q2

qk jq1 cyn s exp Ž i Ž k a y Ž j q 1 . l . j . , 1, 0

if lra is rational, s~

1y

`

kj

Ý

Ý

js0 ksk jy1 q2

c1,k 0 s jq1 exp Ž i Ž k a y Ž j q 1 . l . j . ,

if lra is irrational and < c1 < b < cy1 < - < c 0 < bq1 , 1q

¢

`

t jq1 y1

Ý Ý

js1 ts t jq1

t yjq1 cy exp Ž i Ž yta q Ž j y 1 . l . j . , 1, 0 s

if lra is irrational and < c1 < b < cy1 < ) < c 0 < bq1 .

This gives Gy by Ž6.13.. It remains to observe that the formulas of G " obtained under the assumption < s < - 1 are also valid when < s < G 1. This can be easily checked by direct multiplication. Let us finally show that the conditions Ž6.4. and Ž6.5. are necessary for the canonical AP factorizability of G.

549

CORONA THEOREM AND FACTORIZATION

If b s nra is rational, the matrix function G has a canonical AP factorization only simultaneously with G0 Ž j . s

yi k j

cy1 e

einj y c 0 q c1 e i m j

0 , eyi n j

Ž 6.14.

where n s k q m and nra s krm, because G 0 Ž j . s GŽ m jra .. It follows from Lemma 4.2 in w5x Žsee also w14x. that the matrix-valued function Ž6.14. has a canonical AP factorization if and only if n

det Ž cy1 d iqk , j y c 0 d i , j q c1 d i , mqj . i , js1 / 0,

Ž 6.15.

where d i j is the Kronecker symbol, i.e., n

m Ž y1. Ž c0n y c1k cy1 . / 0.

Thus we have that Ž6.4. is a necessary condition if b s nra is rational. Necessity of Ž6.5. in case of irrational b s nra is proved in w12x. To keep the exposition self-contained, we reproduce it here. Let < c1 < b < cy1 < s < c 0 < bq1 . If G has a canonical AP factorization, then with the help of a small perturbation of c1 , cy1 we get a matrix function of the same form for which Ž6.5. holds. If Ž6.5. is satisfied, then Ž6.6. ] Ž6.9. give that

¡ yc

y1 0

0 dŽ G. s

~

yc0 s 1

0 y1 s yc0 0

cy1 0 , 0

if < c1 < b < cy1 < - < c 0 < bq1 , 0 y1

¢

ys yc0

s

ys yc0

yc0 s 1

ycy1 cy1 y1 1 s 0 y2 c 0

0 ycy1 y1 c1

,

if < c1 < b < cy1 < ) < c 0 < bq1 .

The mapping G ¬ dŽ G . acts continuously from the set of canonically AP factorizable APW matrix functions to the set of constant complex matrices Žsee, e.g., Theorem 7 and Corollary 8 in w8x.. Since for the perturbed matrix functions considered we may obtain both possibilities for dŽ G ., we get a contradiction. Let us remark again that in Theorem 5.1 of w12x it is proved that G, defined by Ž6.1., is not AP factorable at all if < c1 < b < cy1 < / < c0 < 1q b and b s nra is irrational. Hence this condition gives not only a criterion of canonical AP factorizability, but also a criterion of AP factorizability with

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any partial AP indices. In the case of rational b , G is always AP factorizable, but it can happen Žsee, e.g., Theorems 3.1 and 3.2 in w13x. that G is AP factorizable with nonzero partial AP indices if Ž6.4. is violated. Finally, note that with the help of Theorem 6.1 we may construct explicit canonical AP factorizations for all triangular 2 = 2 matrix functions considered in w2x. REFERENCES 1. M. A. Bastos, Yu. I. Karlovich, A. F. dos Santos, and P. M. Tishin, The corona theorem and the existence of canonical factorization of triangular AP matrix functions, 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, A. F. dos Santos, and R. Duduchava, Finite interval convolution operators on the Bessel potential spaces H ps , Math. Nachr. 173 Ž1995., 49]63. 4. L. Childs, ‘‘A Concrete Introduction to Higher Algebra,’’ Springer-Verlag, New York, 1979. 5. I. Feldman, I. Gohberg, and N. Krupnik, On explicit factorization and applications, Integral Equations Operator Theory 21 Ž1995., 430]459. 6. J. B. Garnett, ‘‘Bounded Analytic Functions,’’ Academic Press, New York, 1981. 7. M. Gelfand and I. M. Spitkovsky, Almost periodic factorization: Applicability of the division algorithm, forthcoming. 8. Yu. I. Karlovich, On the Haseman problem, Demonstratio Math. 26 Ž1993., 581]595. 9. Yu. I. Karlovich, The continuous invertibility of functional operators in Banach spaces, Dissertationes Math. 340 Ž1995., 115]136. 10. 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. 11. 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. ¨ 12. Yu. I. Karlovich and I. M. Spitkovsky, Factorization of almost periodic matrix functions, J. Math. Anal. Appl. 193 Ž1995., 209]232. 13. Yu. I. Karlovich and I. M. Spitkovsky, Almost periodic factorization: An analogue of Chebotarev’s algorithm, Contemp. Math. 189 Ž1995., 327]352. 14. N. Krupnik and I. Feldman, On the relation between factorization and inversion of finite Toeplitz matrices, Iz¨ . Akad. Nauk Mold. SSR, Ser. Fiz.-Tekh. Math. 3 Ž1985., 20]26 Žin Russian.. 15. S. Lang, ‘‘Algebra,’’ 3rd ed., Addison-Wesley, Reading, MA, 1993. 16. D. Quint, L. Rodman, and I. M. Spitkovsky, New cases of almost periodic factorization of triangular matrix function, Michigan Math. J., to appear. 17. L. Rodman and I. M. Spitkovsky, Almost periodic factorization and corona theorem, Indiana Uni¨ ersity Math. J., to appear. 18. I. M. Spitkovsky, Factorization of almost-periodic matrix-valued functions, Math. Notes 45 Ž1989., 482]488. 19. J. Xia, Conditional expectations and the corona problem of ergodic Hardy spaces, J. Funct. Anal. 64 Ž1985., 251]274.