c Allerton Press, Inc., 2007. ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2007, Vol. 42, No. 6, pp. 303–319.  c H. A. Amirjanyan, A. G. Kamalyan, 2007, published in Izvestiya NAN Armenii. Matematika, 2007, No. 6, pp. 1–24. Original Russian Text 

FUNCTIONAL ANALYSIS

Factorization of Meromorphic Matrix Functions* H. A. Amirjanyan1* and A. G. Kamalyan2** 1

Russian-Armenian University, Yerevan 2

Yerevan State University

Received September 3, 2007

Abstract—The paper considers a class of matrix-functions defined on some contour in the complex plane that have meromorphic continuations in the interior or exterior domain of that contour. These matrix-functions generally do not admit Wiener-Hopf standard factorization. The paper studies the problem of index factorization, which is a version of Wiener-Hopf factorization. Some criterions for index factorization and exact formulas for particular indices are found along with a constructive method which applies the factorization to solution of an explicit, finite system of linear algebraic equations. MSC2000 numbers : 30E25, 45G05 DOI: 10.3103/S1068362307060015 Key words: Factorization, Partial Indices, Toeplitz operator.

1. INTRODUCTION In such topics as boundary value problems for analytic functions, Toeplitz type operators, etc. it is often necessary to represent a matrix-function G of the order n × n defined on a closed contour Γ of the complex plane C, in the form G(t) = G− (t)Λ(t)G−1 + (t),

t ∈ Γ,

(1.1)

where Λ(t) = diag [tκ1 , . . . , tκn ] (κi ∈ Z, i = 1, . . . , n), the matrix-functions G±1 ± respectively are the + boundary values of some matrix-functions analytic in the interior domain Ω ( 0) and in the exterior domain Ω− ( ∞) of Γ and the numbers κi (i = 1, . . . , n) are called partial indices of G. It is assumed that G± satisfies some additional boundary conditions which particularly provide the uniqueness of the partial indices. Among the factorizations (1.1), the most investigated is the Wiener-Hopf factorization, the exact definition of which will be given later. Criterions for existence of the Wiener-Hopf factorization are known for wide classes of matrix-functions (see eg. [1]-[4]). However, for n > 1 a constructive theory of the Wiener-Hopf factorization exists only for rather narrow classes of matrix-functions, especially for analytic and meromorphic matrix-functions. Some necessary and sufficient conditions for factorization of matrix-functions representable as sums of Smirnov and rational matrix-functions, are obtained by I. M. Spitkovskii (see [5] and [4]). The same author also suggested a recursive factorization procedure. The general approach to factorization of analytic operator-functions is given in [6] and is based on the minimal factorization theory of analytic operator-functions developed by the authors of [6]. Explicit factorizations for different classes of meromorphic matrix-functions are constructed in [7–9]. In spite of methodological differences, the results of that work are stated in terms of power moments (with respect to the contour Γ) of the matrix-function which is the inversion of that to be factorized. A more general factorization than (1.1) is considered in [10, 11] in connection with some Riemann problem. * **

E-mail: [email protected] E-mail: [email protected]

303

304

AMIRJANYAN, KAMALYAN

In [13] (see also [12]), the notion of index factorization is studied leading to an essential extension of the class of matrix-functions admitting the representation (1.1) under weaker assumptions as regards the boundary values of G−1 ± . We note that if the matrix-function G admits Wiener-Hopf factorization, then its index factorization exists and coincides with Wiener-Hopf factorization. The present paper is devoted to investigation of the index factorization of meromorphic matrixfunctions. 2. NOTATION AND PRELIMINARY RESULTS 2.1. Throughout this paper, Ωj (j = 0, 1, . . . , m) are some simply connected, bounded domains in the complex plane C, such that Ωj ⊂ Ω0 , Ωj ∩ Ωi = ∅ (i, j = 1, . . . , m) and the boundaries Γj= ∂Ωj (j = 0, . . . , m) are some closed, rectifiable Jordan curves. It is assumed that the contour Γ = m j=0 Γj m + is oriented so that a motion in positive direction leaves the interior domain Ω = Ω0 \ j=1 Ωj on the +

left-hand side and the exterior domain Ω− = C \ Ω in the right-hand side. Further, by Ep± (0 < p ≤ ∞) we denote the Smirnov classes of the domains Ω± (for definition, ◦

− see [14]-[16] and also [4]), by E − p we denote the class of those functions in Ep , which vanish at the ◦  will stand for the set of all functions from Lp (= Lp (Γ)), that coincide with the infinity, L± L− p p ◦  almost everywhere on Γ. By R we denote angular boundary values of some functions of Ep± E − p

the set of rational functions, by R0 the set of rational functions, whose poles lie outside the contour p− ) we denote the set of functions ϕ meromorphic in Ω+ (Ω− ) possessing p+ (or M Γ. Besides, by M   non-zero polynomials q+ (or non-zero polynomials q− and integers k (k ≥ 0)) such that q+ ϕ ∈ L+ p  ◦  −k − τ0 q− ϕ ∈L p . We use the notation · ◦

− Lp := L+ p +L p ,

Mp± := L± p + R0 ,

L± =



L± p,

M± =

p>0



Mp± ,

p>0

and write Γ ∈ S if all Ωj and C \ Ωj (j = 0, . . . , m) are Smirnov domains (for definition of Smirnov domains and sufficient conditions for Γ ∈ S, see [14]-[18] and [4]). Everywhere below, by X n and X n×n correspondingly we denote the space of n-order vector-columns and n × n-order matrices with elements from the linear space X. Besides, the abbreviations VF and MF will be used for vector-function and matrix-function. The term Wiener-Hopf right factorization in the space Lp (1 < p < ∞) with respect to the contour Γ (or briefly W H(r, p)-factorization) of a MF G will mean (see [4]) the representation (1.1), where  n  ± n , G−1 , q = p/(p − 1) and G± ∈ L± ± ∈ Lq p Λ(t) = diag [tκ1 , . . . , tκn ] ,

t ∈ Γ, κ1 ≤ κ2 ≤ · · · ≤ κn , κi ∈ Z, i = 1, . . . , n.

The numbers κ1 , . . . , κn will be called right partial indices or W H(r, p) partial indices of G. 2.2. In [13], the notions of (r+ , p)-partial indices and (r+ , p)-index factorization of a MF G (or, which is the same, I(r+ , p)-factorization (1 ≤ p ≤ ∞)) were studied. We give the definitions of these and some related notions. Assuming that G is an n × n-order MF whose elements take finite complex values almost  neverywhere such that on the contour Γ, by Dp+ (G) (1 ≤ p ≤ ∞) we denote the space of all VF ϕ+ ∈ L+ p  − n n − n Gϕ+ ∈ Lp , and by Dp (G) the space of all VF ϕ− ∈ Lp for which there are some ϕ ∈ Lp such that ϕ− = Gϕ. ◦

◦

− − By P+ or P− we denote the operators that project Lp to L+ p or to L p (1 ≤ p ≤ ∞) parallel to L p or + Lp respectively, the action of the projectors P± on VF and MF being component-wise.

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

305

Further, we assume that Tp (G) is a Toeplitz operator (1 ≤ p ≤ ∞) defined in Dp+ (G) by the formula Tp (G)ϕ+ = P+ (Gϕ+ ),

τα

(α ∈ C)

and τα− (α ∈ C \ {0}) are the shift operators defined for f ∈ VF or MF by the formulas − τα− = −(α)−1 τα τ0−1 (τ∞ = (τ0 )−1 ).   By Np,j (G) (j ∈ Z) we denote the kernels of the operator Tp τ0−j G . Following [12], the subspace

(τα f )(t) = (t − α)f (t) and

p,j (G) = Np,j (G) + τ0 Np,j (G) N is called (p, j)+ -heredity space and any of direct completions Mp,j (G) in Np,j+1 we call (p, j)+ -index subspace of MF G. We put μp (G, j) = dim Mp,j (G). In [12] it is proved that n (G) :=



μp (G, j) ≤ n.

j∈Z

If n (G) = n, then we say that the MF G admits a finite (r+ , p)-indexation. We suppose that {η1 , . . . , ηs } = {j; μp (G, j) > 0} ,

η1 < η2 < · · · < ηs

(s ∈ N )

and nj = μp (G, ηj ). The numbers κ1 = · · · = κn1 = η1 and κn1 +1 = · · · = κn1 +n2 = η2 , . . . , κn1 +n2 +···ns−1 +1 = · · · = κn (G) = ηs we call the (r+ , p)-partial indices of the MF G. For z ∈ Ω+ we consider also the subspaces Np,j (G; z) = {ϕ(z) : ϕ ∈ Np,j (G)}

and

Mp,j (G; z) = {ϕ(z) : ϕ ∈ Mp,j (G)} .

In [12], it is proved that the dim Np,j (G; z) is a constant for all j ≤ η1 and almost all z ∈ Ω+ . We denote this constant by μp (G; −∞). In the same paper (see also [13]), it is proved that dim Np,j (G; z) is equal (G) for all j > ηs and almost all z ∈ Ω+ and that dim Np,j (G; z) does not exceed to μp (G; −∞) + n (G) for all remaining z. We set μp (G; −∞) + n (G) − μp (G; −∞). μp (G; +∞) := n − n Hence, the below statements (see also [12], [13]). I. μp (G; −∞) = 0 if and only if there exists j ∈ Z such that dim Np,j (G) = 0. II. μp (G; +∞) = 0 if and only if there exist some j ∈ Z and z ∈ Ω+ such that dim Np,j (G, z) = n. In [13], it is proved that if μp (G, −∞) = 0, then

(k − j) μp (j, G), dim Np,k (G) =

k ∈ Z.

j
Hence, by a standard argument (see, eg. [9]), the next statement is true. III. Let μp (G, −∞) = 0, while η0 and ηs+1 be arbitrary integers satisfying η0 ≤ η1 , ηs+1 ≥ ηs , (G) the (r+ , p)-partial indices of the MF G satisfy η0 < ηs+1 . Then for any i = 1, 2, . . ., n  (2.1) κi = η0 + card j : dim Np,j (G) − dim Np,j−1 (G) < i, j = η0 + 1, . . ., ηs+1 . We say that an MF G admits an (r+ , p)-index factorization (see [13]) if G has the representation (1.1) and the factors G± and Λ satisfy the conditions: JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

No. 6

2007

306



 ± n×n

a) G± ∈ Lp

AMIRJANYAN, KAMALYAN

,

+

b) for any z ∈ Ω and V ∈ C n , V = 0, the VF τz−1 G+ V does not belong to Dp+ (G), −1

c) for any z ∈ Ω− and V ∈ C n , V = 0, the VF (τz− )

G− V does not belong to Dp− (G),

d) Λ(t) = diag [tκ1 , . . . , tκn ], where κ1 ≤ · · · ≤ κn are some integers. Also, we shall use the following three statements form [12] and [13]. IV. Let an MF G have finite (r+ , p)-indexation, {η1 , . . . , ηs } = {j, μp (G, j) > 0} (s ∈ N ), and κ1 ≤ · · · ≤ κn be the (r+ , p)-partial indices of G. Then G has an I(r+ , p) factorization. If Mp,ηk (G) (k = 1, . . . , s) are any (p, ηk )+ -index subspaces of G and the columns of the MF ·

·

G+ form a base of the space Mp,η1 (G) + · · · + Mp,ηs (G), then the representation  −1 , where Λ = diag [tκ1 , . . . , tκn ] and G− = GG+ Λ−1 , G = G− Λ G+ is an I(r+ , p) factorization of the MF G. V. Let μp (G, −∞) = 0 and the representation G = G− ΛG−1 + be an I(r+ , p) factorization of the MF G with Λ(t) = diag [tκ1 , . . . , tκn ]. Then G admits a finite (r+ , p)-indexation with (r+ , p) partial indices equal to κ1 , . . . , κn . If {η1 , . . . , ηs } = {j, μp (G, j) > 0} (s ∈ N ),

nj = μp (G, ηj ) (j = 1, . . . , s)

+ + + + , . . . , G+ and G+ 1n1 , G21 , . . . , G2n2 , . . . , Gs1 , . . . , Gsns are the columns of

11 + span G+ k1 , . . . , Gknk (k = 1, . . . , s) is a (p, ηk )+ -index subspace of G.

the MF G+ , then

VI. If for an MF G, ϕ ∈ Lnp and Gϕ = 0 imply ϕ = 0, then the condition c) of the I(r+ , p) factorization is fulfilled also for z ∈ Γ. 2.3. An MF B meromorphic in the domain Ω+ and having angular boundary values almost everywhere exists a non-zero polynomial g such that the on Γ we call p-acceptable in Ω+ (1 ≤ p ≤ ∞)  ifthere n . We denote the set of p-acceptable in Ω+ MFs by condition ϕ+ ∈ Dp+ (B) implies that gBϕ+ ∈ L+ 1 MFp (Ω+ ). By Fp (Ω+ ) we denote the subset MFp (Ω+ ) that consist of analytic in Ω+ MFs. It is not  n . We note, that any of the difficult to see that if B ∈ Fp (Ω+ ), then ϕ+ ∈ Dp+ (B) implies Bϕ+ ∈ L+ 1 following two conditions is sufficient for the inclusion B ∈ MFp (Ω+ ) (see Theorems 1.11 and 1.25 in [4]): n×n  +  , 1) B ∈ Mq

q = p/(p − 1),

2) Γ ∈ S and B ∈ (M + )n×n . ◦

Given an MF G defined on a contour Γ, by D − p (G) (1 ≤ p ≤ ∞) we denote the space of all VFs  ◦ n such that Gϕ− ∈ Ln1 . A meromorphic in Ω− MF G, which has the boundary values almost ϕ− ∈ L − p everywhere on Γ, we call p-acceptable in Ω− if there exist a non-zero polynomial g and a number k ∈ Z  ◦ n ◦ −k − (G) imply τ gGϕ ∈ (k ≥ 0), such that ϕ− ∈D − L 1 . By MFp (Ω− ) we denote the set of all p− p 0 acceptable in Ω− MFs. And by Fp (Ω− ) the subset MFp (Ω− ) consisting of all analytic in Ω− MFs. If  ◦ n ◦ − (G), then Gϕ ∈ G ∈ Fp (Ω− ) and ϕ− ∈D − L 1 . We note, that any of the following two condition is − p sufficient for the inclusion G ∈ MFp (Ω− ) (see Theorems 1.11 and 1.25 in [4]). JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

 n×n q− 1) G ∈ M , q = p/(p − 1),

307

2) Γ ∈ S and G ∈ (M − )n×n . It is not difficult to see that g ∈ R and G ∈ MFp (Ω± ) imply the inclusion gG ∈ MFp (Ω± ). An MF G is meromorphic in the domain Ω± and has angular boundary values almost everywhere on Γ we call regular in Ω± if det G is not identically zero in Ω± . Under the assumption that B(A) is a regular in Ω+ (Ω− ) MF, we consider the following Hankel ◦

−1 operators Hp+ (B −1 ) and Hp− (A−1 ) respectively in Dp+ (B −1 ) and D − p (A ):

Hp+ (B −1 )ϕ = P− (B −1 ϕ) and We set Js =

 s−1

Hp− (A−1 )ϕ = P+ (A−1 ϕ). 

yk z k ; yk ∈ Cn , z ∈ C ,

s ∈ N,

k=0

Js =

 −1

 yk z ; yk ∈ C , z ∈ C , k

n

−s ∈ N,

s

the isomorphisms Ψs : Cn|s| → Js being given by the formulas Ψs y =

s−1

yk z k ,

k=0



T where y = y0T , . . . , ys−1

T

, yk ∈ C n (k = 0, . . . , s − 1), s ∈ N , and Ψs y =

−1

yk z k ,

k=s



 T T

T ,...,y where y = y−1 s

, yk ∈ C n (k = s, . . . , −1), −s ∈ N .

In the present paper we investigate the I(r+ , p)-factorization problem for MF’s from the class MFp (Ω+ ) and MF’s, with inversions belonging to MFp (Ω− ) (1 ≤ p ≤ ∞). + ) be an MF regular in Ω+ . Then N (B) = {0} for s ≤ 0 and Proposition 1. Let p,s  B ∈ Fp (Ω   −1 + −1  Np,s (B) = B ker Hp (B ) J for s > 0. s

Proof: If ϕ+ ∈ Np,s (B), then by standard argument s ≤ 0 and Bϕ+ ∈ Js for s > 0. Besides,  ϕ+ = 0 for   + −1 −1 + −1  Hp (B )Bϕ+ = 0 and hence Np,s (B) ⊂ B ker Hp (B ) J (s > 0). s     n , then obviously ψ+ ∈ If ϕ+ ∈ ker Hp+ (B −1 )J (s > 0), i.e. ϕ+ ∈ Js and ψ+ = B −1 ϕ+ ∈ L+ p s     −s −s Dp+ (B) and Tp τ0 B ψ+ = P+ τ0 ϕ+ = 0.  n×n and A−1 ∈ MF1 (Ω− ), then Proposition 2. If A ∈ L− p   −   j− Np,j (A) = τ0 ker Hp+ τ0j In 



 +  Im Hp− τ0j A−1

,

where j − = 12 (j − |j|), j + = 12 (j + |j|) (j ∈ Z) and In is the unit matrix of the order n × n. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

No. 6

2007

308

AMIRJANYAN, KAMALYAN

 ◦ n + Proof: If ϕ+ ∈ Np,j (A), then there exists some ϕ− ∈ L − such that τ0−j Aϕ+ = ϕ− , i.e. τ0j A−1 ϕ− = p  +   +   n − −j − j −1 ϕ − τ j A−1 ϕ . On the other hand, . Hence τ ϕ = P A τ = H τ0−j ϕ+ ∈ L+ + + − − p p 0 0 0  −   − Hp+ τ0j In τ0−j ϕ+ = 0. To prove the converse statement, suppose  ϕ+ ∈ ker



 −   τ0j In 

Hp+

 +  Im Hp− τ0j A−1

.

 ◦ n  +  ◦ + j − −1 and ϕ ∈ such that τ0j A−1 ψ− = ϕ+ + ϕ− . Then there exist some VFs ψ− ∈ D − L1 − p τ0 A +

But A−1 ∈ MF1 (Ω− ), and therefore there is some polynomial g such that gϕ+ = gτ0j A−1 ψ− − gϕ− ∈  + n  − n L1 ∩ M1 , where the poles of the VF gϕ+ are at infinity, i.e. gϕ+ is a vector-polynomial and  n×n + , and the right-hand side of the equality τ0−j Aϕ+ = hence ϕ+ ∈ R0n . Besides, ϕ+ ∈ R0n , A ∈ L− p   n ◦ + + . On the ψ− − τ0−j Aϕ− is analytic in Ω− and vanishes at infinity. Consequently τ0−j Aϕ+ ∈ L − p   − − − n and Aτ0j ϕ+ ∈ Lnp , i.e. τ0j ϕ+ ∈ Dp+ (A). Therefore, other hand, τ0j ϕ+ ∈ L+ p  −    + Tp τ0−j A τ0j ϕ+ = P+ τ0−j Aϕ+ = 0, and the proof is complete. 2.4. For any VF or MF from L1 , we consider  1 t−k−1 f (t)dt, k ∈ Z. f k = 2πi Γ  n×n     and P− B −1 = 0, then the MF P− B −1 admits If B −1 ∈ Mp+   −1 P− B −1 = Pr Q−1 r = Q P , where Qr and Q are matrix polynomials with unit matrix In elder coefficients, while Pr and P are matrix polynomials satisfying the conditions deg Pr < deg Qr = νr and deg P < deg Q = ν . It is obvious that the above representation is not unique. By νr (B −1 ) and ν (B −1 ) respectively we denote  the least −1 numbers νr and ν , for which such representations are possible. The representations P− B = Pr Q−1 r  −1  (r) (r) −1 n×n and P− B = Q P respectively are equivalent to the existence of matrices Q0 , . . . , Qνr −1 ∈ C ()

()

and Q0 , . . . , Qν −1 ∈ Cn×n satisfying B

−1

−(νr +m) +

ν

r −1

B −1 −(m+k) Qk = 0,

m ∈ N,

(2.2)

Qk B −1 −(m+k) = 0,

m ∈ N.

(2.3)

(r)

k=0

B −1 −(ν +m) + B −1



()

k=0

= 0, we assume that ν (B −1 ) = νr (B −1 ) = 0.  + n×n ∈ M1 , then by Hs+ (s ∈ N ) we denote the Hankel matrix ⎡ −1 B −1 −2 . . . B −1 −s ⎢ B −1 ⎢ −1 ⎢ B −2 B −1 −3 . . . B −1 −s−1 + Hs = ⎢ ⎢ ⎢ ... ... ... ... ⎣

In case P− If B −1



ν

 −1

⎤

B −1 −ν (B−1 ) B −1 −ν (B−1 )−1 . . . B −1 −ν (B−1 )−s+1

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

309

+ + + −1 Let now h+ s = rang Hs for s > 0 and h0 = 0. Then by (2.2), hs = hνr (B−1 ) for s > νr (B ).  n×n       If A−1 ∈ M1− and P+ A−1 = 0, then the MF P+ A−1 admits a representation P+ A−1 = −1 Pr Q−1 r = Q P , where Qr , Q , Pr and P are matrix polynomials satisfying the relations Qr (0) = Q (0) = In , deg Pr ≤ deg Qr ≤ νr and deg P ≤ deg Q ≤ ν . Again we write νr (A−1 ), ν (A−1 ) for the least numbers νr , ν for which this representations are possible. Such representations for P+ (A−1 ) are (r) (r) () () equivalent to the existence of matrices Q1 . . . , Qνr ∈ Cn×n and Q1 . . . , Qν ∈ Cn×n satisfying

A−1 νr +m +

νr

(r) A−1 νr +m−k Qk = 0,

m ∈ N,

(2.4)

m ∈ N.

(2.5)

k=1

A−1 ν +m +

ν

Qk A−1 ν +m−k = 0, ()

k=1

For the case where P+ (A−1 ) = 0, we assume that νr (A−1 ) = ν (A−1 ) = 0.  n×n , then In [19], it is proved that if A−1 ∈ MF1 (Ω− ) ∩ M1−        j j Im H1− τ0 A−1 = Im H1− τ0 A−1  , j ∈ Z, j ≥ 0. 

 − n×n

J−(νr (A−1 )+j)

, then Proposition 3. If A−1 ∈ MF1 (Ω− ) ∩ M1       Im Hp− τ0j A−1 = Im Hp− τ0j A−1 

J−(νr (A−1 )+j)

 ,

j ∈ Z,

 n×n , then by Hs− (−s ∈ N ) we denote the Hankel matrix If A−1 ∈ M1− ⎡ −1 −1 . . . A−1 νr (A−1 ) ⎢ A 1 A 2 ⎢ −1 ⎢ A 2 A−1 3 . . . A−1 νr (A−1 )+1 − Hs = ⎢ ⎢ ⎢ ... ... ... ... ⎣

j ≥ 0.

⎤

A−1 −s A−1 −s+1 . . . A−1 −s+νr (A−1 )−1 .

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

− − We put h− s (s ∈ Z, s ≤ 0) and consider the spaces Fj (j ≤ 0) as follows: hs = rang Hs for s < 0 and −1 − − − −1 h0 = 0, Fj = ker Hj for j < 0 and F0 = C nνr (A ) . Then by (2.5) h− s = h−ν (A−1 ) when s < −ν (A ). 

− matrices B −1 , A−1 and the integration contour. We It is obvious that the numbers h+ s , hs depend on  the −1  + −1 − denote these numbers by hs Γ, B and hs Γ, A respectively.

3. FACTORIZATION OF MATRIX-FUNCTIONS WHICH ARE MEROMORPHIC IN THE INTERIOR DOMAIN OF THE CONTOUR 3.1. We start with a I(r+ , p)-factorization criterion for MFs from the class MFp (Ω+ ). Proposition 4. If an MF B ∈ MFp (Ω+ ) admits an I(r+ , p)-factorization, then det B ±1 has finite set of zeros in Ω+ . −1 Proof: Let a representation B = B− ΛB+ be an I(r+ , p)-factorization of B. Then by the uniqueness theorem det B does not vanish almost everywhere in Γ, since det B± = 0 in Ω± . Besides, B ∈ MFp (Ω+ ) and B+ ek ∈ Dp+ (B) (k = 1, . . . , n), where ek = [0, . . . , 1, . . . , 0]T (k = 1, . . . , n) are the basis vectors in  n×n . Now, the VF gB− Λ Cn , and hence there exists a nonzero polynomial g such that gBB+ ∈ L+ 1  − n×n and can have a pole only at infinity. Therefore, the equality gBB+ = gB− Λ implies belongs to Mp  + n×n  − n×n ∩ Mp . The last inclusion means that gBB+ is a nonzero polynomial matrix. gBB+ ∈ L1

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

No. 6

2007

310

AMIRJANYAN, KAMALYAN

Proposition 5. Let B be a regular MF in Ω+ . Then: 1) if B ∈ MFp (Ω+ ) (1 ≤ p ≤ ∞), then μp (B, −∞) = 0,  n×n p+ 2) if B −1 ∈ M , then μp (B, +∞) = 0.

Proof: Suppose ϕ+ ∈ Dp+ (B) and observe that the MF B is of MFp (Ω+ ), and hence there exists a  n . If j ≤ −deg g and ϕ+ ∈ Np,j (B), then there exists some nonzero polynomial g such that gBϕ+ ∈ L+ 1  ◦ n such that gBϕ+ = τ0j gϕ− . Hence gBϕ+ = 0, and we get ϕ+ = 0 by the regularity of the ϕ− ∈ L − p MF B. Thus, Np,j (B) = {0}, and 1) follows from the above statement I.  n×n  n×n −1 +  . Then there exists a nonzero polynomial q such that qB −1 ∈ L+ . BeNow let B ∈ Mp p sides, one can see that qB −1 ek ∈ Dp+ (B) (k = 1, . . . , n). If j > deg q, then the equality τ0−j BB −1 qek =

τ0−j qek (k = 1, . . . , n) implies B −1 qek ∈ Np,j (B). On the other hand, the regularity of B implies the existence of a point z ∈ Ω+ such that dim Np,j (B, z) = n. Thus, 2) follows from the above statement II. Now, we are ready to prove Theorem 1. If B ∈ MFp (Ω+ ), then B admits an I(r+ , p)-factorization if and only if B −1 ∈ n×n  + . M p

 n×n −1 Proof: If B admits an I(r+ , p)-factorization, then B = B− ΛB+ , where B− ∈ L− and B+ ∈ p  + n×n , Λ(t) = diag [tκ1 , . . . , tκn ] (κi ∈ Z, i = 1, . . . , n). Hence, it is obvious that B is a regular MF Lp  n×n , and consequently in Ω+ . Besides, there is a nonzero polynomial g such that gBB+ = gB− Λ ∈ L+ 1  n×n p+ . The conB− ∈ R0n×n . Further, the representation B −1 = B+ (B− Λ)−1 implies that B −1 ∈ M verse statement follows from Proposition 5 and the above statement IV.  n×n −1 and the representation B− ΛB+ is an I(r+ , p)Corollary 1. If B ∈ MFp (Ω+ ), B −1 ∈ Mp+ factorization of the MF B, then det B− (z) = 0, z ∈ Γ.

Proof: Suppose there exists some z0 ∈ Γ such that det B− (z0 ) = 0. Then B− (z) is a rational MF  ◦ n such that the represenand there exist a non-zero vector V ∈ C n and a rational VF ψ− L − p tation B (z)V = (z − z0 )ψ− (z) is true in a small enough neighborhood of the point z0 . Hence  −1  − −1 −−1 B B− V = B −1 (−z0 )τ0 ψ− ∈ (Lp )n , i.e. τz−0 B− V ∈ Dp− (B), in contradiction to the conτ z0 dition c) of I(r+ , p)-factorization by IV. 3.2. Below we give some explicit formulas for (r+ , p)-partial indices of B ∈ MFp (Ω+ ) in case B −1 ∈  + n×n . Mp   n×n    , s ∈ N and νr (B −1 ) > 0. Then the VF ϕ belongs to ker Hp+ B −1 J Proposition 6. Let B −1 ∈ Mp+ + if and only if the vector Ψ−1 s ϕ belongs to KerHs . In particular,     dim ker Hp+ B −1 J = ns − h+ s . s

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

s

(3.1) Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

 +

Proof: If ϕ ∈ Js , then the equality Hp+ (B −1 )ϕ = Hp P− B Hp+ (B −1 )ϕ =

−1

zm

m=−∞

 −1

311

ϕ |z| great enough implies

s−1

B −1 m−k ϕ k .

(3.2)

k=0

 n×n    . Therefore by (3.2), ϕ ∈ ker Hp+ B −1 J It is not difficult to see that if ϕ ∈ Js , then B −1 ϕ ∈ Mp+ s  T T T ϕ = ϕ , . . . ϕ satisfies the infinite system of equations: if and only if the vector Ψ−1 s 0 s−1 

s−1

B −1 m−k ϕ k = 0,

m = −1, −2 . . .

(3.3)

k=0 + The equalities (3.3) can be true if and only if the vector Ψ−1 s ϕ belongs to Ker Hs , since there exist () some matrices Qi ∈ C n×n (i = 0, 1, . . . , ν (B −1 ) − 1) satisfying (2.3). Besides, (3.1) follows from the obvious equality     dim Ker Hp+ B −1 J = dim KerHs+ . s

 n×n , then the (r+ , p)-partial indices of the MF B for Theorem 2. If B ∈ Fp (Ω+ ) and B −1 ∈ Mp+ −1 νr (B ) = 0 can be determined by

 + −1 − h < i; j = 1, 2, . . . , ν (B ) , i = 1, . . . , n, κi = card j | n + h+ r j−1 j and κ1 = · · · = κn = 0 for νr (B −1 ) = 0.     Proof: For νr B −1 = 0, the proof is obvious. If νr B −1 = 0, the proof follows from Propositions 1, 5, 6 and the equality (2.1). 3.3. Now, we proceed to the construction of the factorization of the MF B assuming that B ∈ Fp (Ω+ )  n×n . and B −1 ∈ Mp+ By ωs1 , ωs2 : C ns → C n(s+1) (s ∈ N ) we denote the matrix operators: ⎞ ⎛ ⎛ ⎜ In 0 . . . 0 ⎟ ⎜ 0 0 ... ⎟ ⎜ ⎜ ⎜ 0 In . . . 0 ⎟ ⎜ In 0 . . . ⎟ ⎜ ⎜ ⎟ ⎜. ⎜ ωs2 = ⎜ 0 In . . . ωs1 = ⎜ .. ⎟, ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ .. ⎜ 0 0 . . . In ⎟ ⎜. ⎠ ⎝ ⎝ 0 0 ... 0 0 0 ...

⎞ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ ⎠ In

+ (i = 1, 2, s ∈ N ). Then obviously ωsi Ker Hs+ ⊂ Ker Hs+1

 n×n   , ν B −1 > 0, θ0 = Ker H1+ and θs , s ∈ N be Proposition 7. Let B ∈ Fp (Ω+ ), B −1 ∈ Mp+ some direct completion of ωs1 Ker Hs+ + ωs2 Ker Hs+ + in Ker Hs+1 . Then

  p,s (B) = B −1 Ψs+1 ω 1 Ker H+ + ω 2 Ker H+ N s s s s

(3.4)

and Ms−1 := B −1 Ψs θs−1 is a (p, s − 1)+ -index subspace of MF B, such that dim Ms−1 = dim θs−1 (s ∈ N ). JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

No. 6

2007

312

AMIRJANYAN, KAMALYAN

Proof: In view of Propositions 1 and 6, Np,s (B) = {0} for s ≤ 0 and Np,s (B) = B −1 Ψs Ker Hs+ for s > 0. Therefore, (3.4) follows from the easily verifiable equalities Ψs Ker Hs+ = Ψs+1 ωs1 Ker Hs+

and

τ0 Ψs Ker Hs+ = Ψs+1 ωs2 Ker Hs+ .

p,0 (B) = 0 implies that M0 is a (p, 0)+ -index subspace. Besides, N p,s(B) ∩ Ms . Then there exist some vectors y1 , y2 ∈ Ker Hs+ and y ∈ θs , such that Now let f ∈ N f = B −1 Ψs+1 ωs1 y1 + B −1 Ψs+1 ωs2 y2 = B −1 Ψs+1 y.   Hence B −1 Ψs+1 ωs1 y1 + ωs2 y2 − y = 0 and consequently ωs1 y1 + ωs2 y2 = y, i.e. y = 0, implying f = 0. On the other hand, + Np,s+1 (B) = B −1 Ψs+1 Ker Hs+1

= B −1 Ψs+1

#  · $ ωs1 Ker Hs+ + ωs2 Ker Hs+ + θs

= B −1 Ψs+1 ωs1 Ker Hs+ + B −1 Ψs+1 ωs2 Ker Hs+ + B −1 Ψs+1 θs p,s (B) + Ms . =N Thus, obviously dim Ms−1 = dim θs−1 and the proof is complete. The following theorem follows from the above statement IV and Proposition 7.  n×n , ν (B −1 ) > 0 and η1 < · · · < ηs be all possible values Theorem 3. Let B ∈ Fp (Ω+ ), B −1 ∈ Mp+ of κ1 ≤ · · · ≤ κn . Further, let {Xηi ,1 , . . . , Xηi ,ni } be the bases of the spaces θηi (i = 1, . . . , s) and Uηi ,j = B −1 ψηi +1 Xηi ,j ,

i = 1, . . . , s, j = 1, . . . , nj .

Then the conditions B+ = [Uη1 ,1 , . . . , Uη1 ,n1 , Uη2 ,1 . . . , Uηs ,ns ] , Λ = diag [tκ1 , . . . , tκn ]

and

B− = BB+ Λ−1

  −1 imply that the representation B = B− ΛB+ is an I(r+ , p)-factorization of the MF B. If ν B −1 =  −1 is a I(r+ , p)-factorization of the MF B. 0, then the representation B = In In B −1  n×n , then it is not difficult to prove that there exists Remark 1. If B ∈ MFp (Ω+ ) and B −1 ∈ Mp+  n×n . Theorems a polynomial q with zeros in Ω+ , such that qB ∈ Fp (Ω+ ) and (qB)−1 ∈ M+ p 2 and 3 suggest a construction of an I(r+ , p)-factorization of the MF qB. In addition, if the %B % −1 is an I(r+ , p)-factorization of the MF qB, then B = B− ΛB −1 , %− Λ representation qB = B + + −k % k −1 % % where B− = τ0 q B− (k = deg q), Λ = τ0 Λ and B+ = B+ , an I(r+ , p)-factorization of the MF B. In particular if νr q −1 B −1 > 0, then the (r+ , p)-partial indices of the MF B can be determined by the formulas:

    −1 −1 −1 −1 − h+ < i, κi = −deg q + card j | n + h+ j−1 Γ, q B j Γ, q B   j = 1, 2, . . . , νr q −1 B −1 , 

If νr q −1 B

i = 1, . . . , n.

 −1

= 0, then κ1 = · · · = −κn = −deg q.  n×n + , then there exists a polynomial q0 such In addition, if B ∈ MFp (Ω+ ) and B −1 ∈ M p   n×n , and the I(r+ , p)-factorization of the that its zeros lie on the contour Γ and q0 B −1 ∈ Mp+ −1 −1 MF q0 B can be constructed as described above, since q0 B ∈ MFp (Ω+ ). The factorization of MF B can be determined as in [20]. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

313

3.4. There is a simple formula for determining the (r+ , p)-total index κ = κ1 + · · · + κn of a MF B.  n×n Theorem 4. If B ∈ MFp (Ω+ ) and B −1 ∈ Mp+ , then the (r+ , p)-total index of B equals the difference between the numbers of zeros and poles of the function det B in the domain Ω+ , calculated with their multiplicities.

Proof: We denote ε = 2−1

min

i, j = 0, . . . , m j = i

d(Γi , Γj ) and

Uε = {z ∈ C; |z| < ε} .

Let Ωi,ε (i = 0, . . . , m) be simply connected domains in the complex plane with boundaries Γi,ε = ∂Ωi,ε that satisfy Ω0,ε ⊂ Ω0 ,

Ω0 ⊂ Ω0,ε + Uε ,

Ωi ⊂ Ωε,i ,

Ωε,i ⊂ Ωi + Uε ,

i = 1, . . . , m.  Further, assume that Γi,ε are closed, rectifiable Jordan curves and the contour Γε = m j=0 Γj,ε is oriented m + so that moving in positive direction the interior domain Ωε = Ω0,ε \ j=1 Ωj,ε remains on the left. Examples of such domains can be constructed using functions which realize conformal mappings of the domains Ω0 and C \ Ωi (i = 1, . . . , m) to the unit disc. Besides, Γε,i (i = 1, . . . , m) can be chosen to be analytic. For ε > 0 small enough the MF B is analytic and is invertible in some domain ±1 ∈ M + (Γ ), by Theorem 2.6 of [4] the MF B admits a W H(r, p)containing the set Ω+ \ Ω+ ε . Since B ∞ ε −1 % factorization on Γε , that is B = B− ΛB+ . Let Λ = diag [τ0κ1 , . . . , τ0κn ]. Then by κ = κ1 + · · · + κn equals the difference between the numbers of zeros and poles of det B in Ω+ ε . In addition, the MFs ±1 + ±1 n×n − are analytic in Ωε = C \ Ωε and continuous on Γε . Consequently, B− are analytic in B− ∈ R − −1 + %+ , and Ω and continuous on Γ. Defining B+ = B B− Λ, we observe that B+ coincides on Ωε with B  + n×n and satisfies the condition b) see the definition of I(r+ , p)-factorization therefore, B+ ∈ Lp (Γ)  n + for z ∈ Ω . If z ∈ Γ and there exists some V ∈ C n , V = 0, such that τz−1 B+ V ∈ L+ p (Γ) , then Bτz−1 B+ V = τz−1 B− ΛV and det(B− Λ)(z) = 0, hence τz−1 B− ΛV ∈Lnp , implying τz−1 B+ V ∈ Dp+ (B). Thus, the MF B+ satisfies the mentioned condition b) also when z ∈ Γ. Hence, the representation −1 is an I(r+ , p)-factorization of B on the contour Γ. Therefore, the proof of our theorem B = B− ΛB+ follows from the fact that det B does not have zeros in the set Ω+ \ Ω+ ε. 4. FACTORIZATION OF MATRIX-FUNCTIONS WHICH ARE MEROMORPHIC IN THE EXTERIOR DOMAIN OF THE CONTOUR 4.1. The proof of the following statement is similar to the proof of Proposition 4. Proposition 8. If an MF A admits an I(r+ , p)-factorization and A−1 ∈ MFp (Ω− ), then det A±1 possess finite sets of zeros in Ω− . Proposition 9.

n×n  − , then μp (A, +∞) = 0. 1) If A ∈ M p

2) If a MF A is regular in Ω− and A−1 ∈ MFp (Ω− ), then μp (A, −∞) = 0.  n×n p− Proof: If A ∈ M , then there exist a nonzero polynomial q and a number i ∈ Z (i ≥ 0), such that  ◦ n×n . Besides, qek ∈ Dp+ (A) (k = 1, . . . , n). If j ≥ i, then qek ∈ Np,j (A) (k = 1, . . . , n) τ0−i qA ∈ L − p (due to the representation of τ0−j Aqek (k = 1, . . . , n)) and so there exists some z ∈ Ω+ such that dim Np,j (A, z) = n. Thus, 1) follows from the statement II. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

No. 6

2007

314

AMIRJANYAN, KAMALYAN ◦

Now suppose A−1 ∈ MFp (Ω− ). If ϕ− ∈D − (A−1 ), then there exist a polynomial g and a number  ◦ pn . If j ≤ −k and ϕ+ ∈ Np,j (A), then there exists some k ∈ Z (k ≥ 0), such that τ0−k gA−1 ϕ− ∈ L − p  ◦ n ◦ j −1 −1 such that τ0−j Aϕ+ = f− , and it is obvious that f− ∈D − f− ∈ L − p p (A ). Besides, gϕ+ = gτ0 A f−  ◦ n   n and j ≤ −k imply that gϕ+ ∈ L − ∩ L+ , i.e. ϕ+ = 0. Thus, Np,j (A) = {0} (j ≤ −k), and 2) p p follows from the statement I. Theorem 5. If A−1 ∈ MFp (Ω− ), then the MF A admits an I(r+ , p)-factorization if and only if  n×n p− A∈ M .

Proof: Suppose the representation A = A− ΛA−1 + is an I(r+ , p)-factorization of the MF A. Then  ◦ n×n , since there exists a polynomial g and a number m ∈ Z (m ≥ 0) such that τ0−m gA−1 A− ∈ L − 1 and the MF A−1 is p-acceptable in Ω− . The equality gA−1 A− Λ = gA+ A−1 A− = A+ Λ−1 ∈ Ln×n 1  + n×n  − n×n implies gA+ ∈ Lp ∩ Mp , where the MF gA+ can have a pole only at infinity. Thus, the MF  n×n p− . The converse statement follows from Proposition 9 gA+ is a polynomial matrix. Hence A ∈ M and IV.  n×n and the representation A = A− ΛA−1 Corollary 2. If A−1 ∈ MFp (Ω− ), A ∈ Mp− + is an I(r+ , p)-factorization of the MF A, then det A+ (z) = 0, z ∈ Γ.

Proof: similar to the proof of Corollary 1. 4.2. Below we give some explicit formulas for (r+ , p)-partial indices of a MF A, valid for the case when  n×n  n×n and A−1 ∈ MF1 (Ω− ) ∩ M1− . A ∈ L− p  n×n ∩ MF1 (Ω− ) and ν (A−1 ) > 0, then Proposition 10. If A−1 ∈ M1− ' &    − h− j ≤ 0, = h− dim ker Hp+ τ0j In  j , − −ν (A−1 ) −1 Im Hp (A



)

j ∈ Z.

Proof: Let j < 0 and f ∈ J−νr (A−1 ) . Then, following [19] one can verify that for |z| small enough 

−1

∞

 zm Hp− (A−1 )f (z) = m=0

A−1 m−k f k .

k=−νr (A−1 )

Consequently −j−1    

z m+j Hp+ τ0j In Hp− (A−1 )f (z) = m=0

Hence, by Proposition 3 &    ker Hp+ τ0j In 

' Im Hp− (A−1 )

−1

k=−νr (A−1 )

 f ∈ F = Hp− (A−1 )f ; Ψ−1 . j −1 −νr (A )

Suppose now that Sj = Ψ−νr (A−1 ) Fj (j < 0) and observe that &     −1    j − +  I = ker H τ Im Hp A  n p 0 S j

A−1 m−k f k .

Im Hp− (A−1 )

' ,

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

    ker Hp− A−1 S = ker j

By these two equalities &    dim ker Hp+ τ0j In 

&

  Hp− A−1 J

' Im Hp− (A−1 )

Hence, in view of

& dim ker

Hp−

−νr (A−1 )

& = dim Sj − dim ker

 −1   A J

Hp−

.

 −1  A J

' −νr (A−1 )

.

(4.1)

' −νr (A−1 )

(4.2)

= dim F−ν (A−1 )

(see [19]) and dim Sj = dim Fj = nνr (A−1 ) − h− j we get ' &    − = nνr (A−1 ) − h− dim ker Hp+ τ0j In  j − dim F−ν (A−1 ) = h−ν − −1 Im Hp (A

315

'

)

 (A

−1 )

− h− j .

It remains to check that for j = 0, the proof follows from (4.2).  n×n    n×n and A−1 ∈ MF1 (Ω− ) ∩ M1− , then for ν A−1 > 0 the Theorem 6. If an MF A ∈ L− p (r+ , p)-partial indices of A can be determined by the formulas

 − −1 − h < i, j = −ν (A ) + 1, . . . , 0 κi = −ν(A−1 ) + card j | h−  j−1 j (i = 1, . . . , n). And κ1 = · · · = κn = 0 for ν (A−1 ) = 0.

Proof: For the case νr (A−1 ) = ν (A−1 ) = 0, the proof is obvious. We have A−1 ∈ MFp (Ω− ). Hence by Proposition 9 the MF A admits a finite (r+ , p)-indexation. The proofs of the remaining statements follow from Propositions 2, 10 and formula (2.1).  n×n and A−1 ∈ 4.3. Now we turn to factorization of the MF A under the conditions that A ∈ L− p  − n×n ∩ MF1 (Ω− ). M1     Let νr A−1 > 0. Then Propositions 2 and 10 imply that Np,j (A) = Im Hp− τ0j A−1 for j ≥ 0 and Np,j (A) = τ0j Hp− (A−1 )Ψ−νr (A−1 ) Fj for j < 0. Given some matrices Q1 , Q2 , . . . , Qνr (A−1 ) ∈ C n×n −1

−1

satisfying the relations (2.4), we consider an operator KQ : C nνr (A ) → C nνr (A ) defined by #  T  T T $T , −Qνr (A−1 )−1 yνr (A−1 ) + y1 · · · −Q1 yνr (A−1 ) + yνr (A−1 )−1 KQ y = −Qνr (A−1 ) yνr (A−1 ) # $T and yi ∈ C n (i = 1, . . . , νr (A−1 )). It is clear that KQ admits the where y = y1T , y2T , . . . , yνTr (A−1 ) matrix representation

⎛

⎜ 0 0 . . . 0 −Qνr (A−1 ) ⎜ ⎜ In 0 . . . 0 −Qνr (A−1 )−1 KQ = ⎜ ⎜ .. ⎜. ⎝ 0 0 . . . In −Q1 .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Observe now that KQ Fj ⊂ Fj+1 (j ≤ −1), and it is not difficult to verify (see also [19]) that Hp− (A−1 )Ψ−νr (A−1 ) KQ y =

∞

m=0

zm

−1

A−1 m−k Ψ−νr (A−1 ) KQ y k

k=−νr (A−1 )

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

No. 6

2007

316

AMIRJANYAN, KAMALYAN

for |z| small enough. By (2.4) we get Hp− (A−1 )Ψ−νr (A−1 ) KQ y

=

∞

m=0

νr (A−1 )

z

m



A−1 m+k+1 yk ,

k=1

hence − y. τ0 Hp− (A−1 )Ψ−νr (A−1 ) KQ y = Hp− (A−1 )Ψ−νr (A−1 ) y − H−1

(4.3)

In addition, for j ≤ −1 Hp− (A−1 )Ψ−νr (A−1 ) Fj = τ0 Hp− (A−1 )Ψ−νr (A−1 ) KQ Fj ,

(4.4)

since Fj ⊂ F−1 (j ≤ −2). To finish the proof, it suffices to suppose that θj (j ≤ −1) is a direct − C n. completion of Fj + KQ Fj in Fj+1 and θ0 is a direct completion of Im H−1  n×n −1  − n×n , A ∈ M1 ∩ MF1 (Ω− ) and νr (A−1 ) = 0. If j < 0, Proposition 11. Let an MF A ∈ L− p then the spaces Mp,j = τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) θj are (p, j)+ -index subspaces of A, and if j = 0, then θ0 is a (p; 0)+ -index subspace of the MF A. Besides, dim Mp,j = dim θj (j < 0).

Proof: If f ∈ Hp− (A−1 )Ψ−νr (A−1 ) (Fj + KQ Fj ) ∩ Hp− (A−1 )Ψ−νr (A−1 ) θj , where j < 0, then there exist some y1 , y2 ∈ Fj and y ∈ θj , such that   f = Hp− (A−1 )Ψ−νr (A−1 ) y = Hp− (A−1 )Ψ−νr (A−1 ) y1 + KQ y2 , i.e. Hp− (A−1 )Ψ−νr (A−1 ) (y − (y1 + KQ y2 )) = 0. Hence, in view of (2.5) one can see that y − y1 − KQ y2 ∈ F−ν (A−1 ) , and therefore y − y1 − KQ y2 ∈ Fj . So, we obtain y ∈ (Fj + KQ Fj ) ∩ θj , i.e. y = 0 and consequently f = 0. On the other hand, by Propositions 2 and 10 we get Np,j+1 (A) = τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) Fj+1 = τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) (Fj + KQ Fj ) + τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) θj   = τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) KQ Fj + τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) Fj + Mp,j . By (4.4), Np,j (A) = τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) KQ Fj , τ0 Np,j (A) = τ0j+1 Hp− (A−1 )Ψ−νr (A−1 ) Fj , and therefore Mp,j (j < 0) are (p, j)+ -index subspaces of the MF A. Now let j = 0. Then it follows that C n ⊂ Np,1 (A) and τ0 Np,0 (A) ⊂ Np,1 (A). On the other hand, n C ∩ τ0 Np,0 (A) = {0}. Besides, by

(k − j)μp (G, j), k, j ∈ Z, dim Np,k (A) = j
(see [13]) we get dim Np,1 (A) = n + κ

(κ = |κ1 + · · · + κn |) and

dim Np,0 (A) = κ.

Hence ·

·

Np,1 (A) = C n + τ0 Np,0 (A) = C n + τ0 Im Hp− (A−1 ). By (4.3), the spaces Im Hp− (A−1 ) + τ0 Im Hp− (A−1 ) and

− τ0 Hp− (A−1 )Ψ−νr (A−1 ) (KQ y + x) + H−1 y,

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

where y and x are arbitrary vectors from C nνr (A Im Hp− (A−1 ) + τ0 Im Hp− (A−1 ), i.e.

−1 )

317

− , coincide. Taking x = −KQ y, we get Im H−1 ⊂ ·

− + τ0 Np,0 (A). Np,0 (A) + τ0 Np,0 (A) = Im Hp− (A−1 ) + τ0 Im Hp− (A−1 ) = Im H−1 ·

·

p,0 (A) + θ0 . To finish the proof, The equality Np,1 (A) = C n + τ0 Im Hp− (A−1 ) implies that Np,1 (A) = N − −1 it remains to note that if y ∈ θj (j < 0) and Hp (A )Ψ−νr (A−1 ) y = 0, then y = 0, i.e. dim Mp,j = dim θj . Proposition 11 and statement IV imply the following theorem.  n×n  n×n −1 , A ∈ MF1 , (Ω− ) ∩ M1− , νr (A−1 ) = 0 and η1 < · · · < Theorem 7. Let an MF A ∈ L− p ηs be all possible values of κ1 ≤ · · · ≤ κn . Further, let {yηi ,1 , . . . , yηi ,ni } (i = 1, . . . , s) be the bases of the spaces θηi (i = 1, . . . , s) and the VFs Uηi ,j (i = 1, . . . , s, j = 1, . . . , ni ) be defined as Uηi ,j = τ0ηi +1 Hp− (A−1 )Ψ−νr (A−1 ) yηi ,j U0,j = y0,j

if

if

ηs = 0,

ηi < 0, i = 1, . . . , s, j = 1, . . . , ni , j = 1, . . . , ns .

Then, under the additional conditions A+ = [Uη1 ,1 , . . . , Uη1 ,n1 , Uη2 ,1 , . . . , Uηs ,ns ] , Λ(t) = diag [tκ1 , . . . , tκn ] ,

A− = AA+ Λ−1 ,

−1 the representation A = A− ΛA−1 + is an I(r+ , p)-factorization of A. If νr (A ) = 0, then the representation A = AIn In is an I(r+ , p)-factorization of A. n  4.4. Consider now the case where A ∈ Mp− (Γ) and A−1 ∈ MFp (Ω− ). We suppose that Ωε,i (i = 0, . . . , m) are some simply connected domains with the boundary 

Γε,i = ∂Ωε,i , for which hold the inclusions Ω0 ⊂ Ωε,0 , Ωε,0 ⊂ Ω0 + Uε , Ωi,ε ⊂ Ωi , Ωi ⊂ Ωi,ε + Uε (i = 1, . . . , m). Besides, we suppose that Γi,ε (i = 0, . . . , m) are closed, rectifiable Jordan curves and that   + the contour Γε = m j=0 Γε,j is oriented so that moving in positive direction the interior domain Ωε =   Ω0,ε \ m j=1 Ωε,j remains on the left. Our next theorem makes it possible to reveal the I(r+ , p)-factorization of the MF A on the contour Γ by the factorization of A on the contour Γε . n×n −1  , A ∈ MFp (Ω− ) and an integer k ≥ 0 and a polynomial q with Theorem 8. Let A ∈ Mp− (Γ)  n×n . Then for ε > 0 small enough the MF τ0−k qA admits zeros in Ω− be such that τ0−k qA ∈ L− p (Γ) %− Λ %A %−1 a W H(r, p)-factorization with respect to the contour Γε . If the representation τ0−k qA = A + −k  n×n % is a W H(r, p)-factorization of the MF τ0 qA with respect to the contour Γε , then A+ ∈ R . If %+ q, Λ = Λτ % k and A− = τ −k qAA %+ Λ % −1 , then the representation A = A− ΛA−1 is an I(r+ , p)A+ = A + 0 0   −k −1 −1 factorization of A on Γ. In particular, for ν τ0 q A > 0 the (r+ , p)-partial indices of A can be calculated by the formulas

  κi = −ν τ0−k q −1 A−1 + k

    

 , τ −k q −1 A−1 − h− Γ , τ −k q −1 A−1 < i , Γ +card j | h− ε 0 ε 0 j−1 j   For ν τ0−k qA−1 = 0, we have κ1 = · · · = κn = k. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

i = 1, . . . , n.

No. 6

2007

318

AMIRJANYAN, KAMALYAN

 n×n Proof: It follows from the conditions A ∈ Mp− (Γ) and A−1 ∈ MFp (Ω− ), that the number ε > 0 can be chosen to render the MF A analytic and det A = 0 in some domain containing the set  ±1 + + −k − (Γ ), and therefore by Theorem 2.7 of [4], τ −k qA admits a Ωε \ Ω . Besides, τ0 qA ∈ M∞ ε 0 W H(r, p)-factorization with respect to the contour Γε . Simultaneously, this factorization is an I(r+ , p)%+ ∈ Rn×n , in essence, is contained in the factorization with respect to the contour Γε . In essence A proof of Theorem 4. Besides, one can verify that the factors A± and Λ satisfy the conditions defining the I(r+ , p)-factorization of a MF with respect to the contour Γ. Formulas for partial indices follow from % The proof is complete. Theorem 6 and the equality Λ = τ0k Λ. The following statement is proved by combining the proofs of Theorem 8 and Theorem 4. n×n  and A−1 ∈ MFp (Ω− ), then the sum of (r+ , p)-partial indices of Theorem 9. If A ∈ Mp− (Γ) the MF A coincides with the difference between the numbers of poles and zeros of the function det A in Ω− , taken with the multiplicities. n×n  p− there p− and A−1 ∈ MFp (Ω− ). Then by the definition of the class M Remark 2. Let A ∈ M  n×n . Consequently exists a polynomial q0 whose zeros lie on the contour Γ, such that q0 A ∈ Mp− −1 − (q0 A) ∈ MFp (Ω ). By Theorems 7 and 8 it is possible to construct an I(r+ , p)-factorization of the MF q0 A. The factorization of MF B can be constructed as in [20]. REFERENCES 1. N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1968). 2. I. N. Vekua, Systems of Singular Integral Equations (Nauka, Moscow, 1970). 3. K. Clancey, I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators (Birkhauser Verlag, Basel, 1981). 4. G. S. Litvinchuk and I. M. Spitkovskii, Factorization of Measurable Matrix Functions (Akademie-Verlag, Berlin, 1987). 5. I. M. Spitkovskii, “Factorization of Measurable Matrix-Functions and Related Theories of Systems of Singular Integral Equations and Riemann Vector Boundary Value Problem”, I, Differential Equations, (4) 697-709 (1981). 6. H. Bart, I. Gohberg and M. A. Kaashoek, “Explicit Wiener-Hopf Factorization and Realization”, in Operator Theory: Advances and Applications, Constructive Methods of Winer-Hopf Factorization 21, 235-316 (1997). 7. I. Gohberg, L. Lerer and L. Rodman, “Factorization Indices for Matrix Polynomials”, Bull. AMS 84, 275-277 (1978). 8. V. M. Adukov, “Wiener-Hopf Factorization of Meromorphic Matrix-Functions”, Algebra and Analysis 4 (1), 54-74 (1992). 9. A. G. Kamalyan, “Generalized Factorization of Bounded Holomorphic Matrix-Functions”, Izv. NAN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)] 32 (2), 19-38 (1997). 10. I. M. Spitkovskii, “Generalized Factorization of Matrix-Functions and Riemann Boundary Value Problem With Infinite Partial Indices”, DAN SSSR [USSR Academy of Sciences, Reports] 286 (13), 559-562 (1986). 11. I. M. Spitkovskii, “On Riemann Boundary Value Vector Problem With Infinite Defect Numbers and the Related factorization of Matrix-Functions”, Math Notes [Matematicheskii Sbornik] 135 (4), 553-550 (1988). 12. A. G. Kamalyan, “Some Properties of Kernels of Toeplitz Operators”, Dokladi NAN Armenii [National Academy of Sciences of Armenia, Reports] (in print) 107 (4), 316-322. 13. A. G. Kamalyan, “Index Factorization of Matrix-Functions”, Dokladi NAN Armenii [National Academy of Sciences of Armenia, Reports] (in print) 108 (1). 14. I. I. Privalov, Boundary Properties of Analytic Functions (GITTL, Moscow, 1950). 15. G. M. Goluzin, Geometrical Theory of Functions of Complex variable (Nauka, Moscow, 1966). 16. G. Ts. Tumarkin and S. Ya. Khavinson, “On Definition of Analytic Functions of Class E in Multiply Connected Domains”, UMN [Advances of Mathematical Sciences] 13 (1), 201-206 (1958). JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

Vol. 42 No. 6 2007

FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS

319

17. B. V. Khvedelidze, “The Method of Cauchy Type Integrals in Discontinuous Boundary Value Problems of the Theory of Holomorphic Functions of Single Complex variable”, in Sovremennie Problemi Matematiki (Itogi Nauki i Tekhniki) [Contemporary Problems of mathematics (Results Science and Technic)] 7, 5-162 (VINITI, Moscow, 1975). 18. E. M. Dinkin and B. P. Osilenker, “Weighted Estimates of Singular Integrals and Their Applications”, in Matematicheskii Analiz (Itogi Nauki i Tekhniki) [Mathematical Analysis (Results Science and Technic)] 21, 42-129 (VINITI, Moscow, 1983). 19. H. A. Amirjanyan, “On Some Properties of Hankel Operator With Meromorphic Symbol”, Matematika v Visshei Shkole [Mathematics in Higher School] 3 (2), (2007). 20. H. A. Amirjanyan, “On Influence of Simplest Factors on Factorization of a Matrix-Function”, Vestnik RAU, Seriya Fiz.-mat. i Estestvenne Nauki [Bulletin of RAU, Series of Physical-Mathematical and Natural Sciences] 2, 12-21 (2007).

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

No. 6

2007