Factorization of generalized γ-generating matrices
The class of γ–generating matrices and its subclasses of regular and singular γ–generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular–singular factorization. The class of generalized γ-generating matrices was...
Збережено в:
| Дата: | 2017 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2017
|
| Назва видання: | Український математичний вісник |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/169379 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Factorization of generalized γ-generating matrices / O. Sukhorukova // Український математичний вісник. — 2017. — Т. 14, № 4. — С. 575-594. — Бібліогр.: 20 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-169379 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1693792020-06-12T01:26:18Z Factorization of generalized γ-generating matrices Sukhorukova, O. The class of γ–generating matrices and its subclasses of regular and singular γ–generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular–singular factorization. The class of generalized γ-generating matrices was introduced in [14,20]. In the present paper subclasses of singular and regular generalized γ-a theorem of existence of regular–singular factorization for rational generalized γ-generating matrix is found. 2017 Article Factorization of generalized γ-generating matrices / O. Sukhorukova // Український математичний вісник. — 2017. — Т. 14, № 4. — С. 575-594. — Бібліогр.: 20 назв. — англ. 1810-3200 http://dspace.nbuv.gov.ua/handle/123456789/169379 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The class of γ–generating matrices and its subclasses of regular and singular γ–generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular–singular factorization. The class of generalized γ-generating matrices was introduced in [14,20]. In the present paper subclasses of singular and regular generalized γ-a theorem of existence of regular–singular factorization for rational generalized γ-generating matrix is found. |
| format |
Article |
| author |
Sukhorukova, O. |
| spellingShingle |
Sukhorukova, O. Factorization of generalized γ-generating matrices Український математичний вісник |
| author_facet |
Sukhorukova, O. |
| author_sort |
Sukhorukova, O. |
| title |
Factorization of generalized γ-generating matrices |
| title_short |
Factorization of generalized γ-generating matrices |
| title_full |
Factorization of generalized γ-generating matrices |
| title_fullStr |
Factorization of generalized γ-generating matrices |
| title_full_unstemmed |
Factorization of generalized γ-generating matrices |
| title_sort |
factorization of generalized γ-generating matrices |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169379 |
| citation_txt |
Factorization of generalized γ-generating matrices / O. Sukhorukova // Український математичний вісник. — 2017. — Т. 14, № 4. — С. 575-594. — Бібліогр.: 20 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT sukhorukovao factorizationofgeneralizedggeneratingmatrices |
| first_indexed |
2025-07-15T04:07:21Z |
| last_indexed |
2025-07-15T04:07:21Z |
| _version_ |
1837684429096484864 |
| fulltext |
Український математичний вiсник
Том 14 (2017), № 4, 575 – 594
Factorization of generalized
γ-generating matrices
Olena Sukhorukova
(Presented by V. O. Derkach)
Abstract. The class of γ–generating matrices and its subclasses of reg-
ular and singular γ–generating matrices were introduced by D. Z. Arov
in [8], where it was shown that every γ-generating matrix admits an es-
sentially unique regular–singular factorization. The class of generalized
γ-generating matrices was introduced in [14, 20]. In the present paper
subclasses of singular and regular generalized γ-generating matrices are
introduced and studied. As the main result of the paper a theorem
of existence of regular–singular factorization for rational generalized γ-
generating matrix is found.
Key words and phrases. γ-generating matrices, J-inner matrix val-
ued function, denominator, associated pair, generalized Schur class, re-
producing kernel space, Potapov–Ginzburg transform, Krĕın–Langer fac-
torization.
1. Introduction
The notion of a γ–generating matrix was introduced by D. Z. Arov
in [8] in connection with the study of completely indeterminate Nehari
problem on the unit circle T (see [1,2,10]), and for a real line R (see [10]).
Let jpq =
[
Ip 0
0 −Iq
]
. We recall that a mvf (matrix valued function)
A =
[
a11 a12
a21 a22
]
, where a11 and a22 are p×p and q×q blocks, respectively,
is called a γ-generating matrix of the class Mr(jpq), if:
(1) A is measurable on R and takes jpq-unitary values for a.e. µ ∈ R;
Received 14.10.2017
This work was supported by a Volkswagen Stiftung grant and grant of the Ministry
of Education and Science of Ukraine (project 0115U000556).
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
576 Factorization of generalized γ-generating matrices
(2) a22(µ) and a∗11(µ) are boundary values of holomorphic mvf’s a22(λ)
and a#11(λ), such that a−1
22 and (a#11)
−1 are outer mvf’s from the
Schur classes Sp×p and Sq×q, respectively;
(3r) s21 := −a−1
22 a21 belongs to the Schur class Sq×p of holomorphic in
C+ with values in the set of contractive mvf’s, i.e. Ip−s(λ)∗s(λ) ≥ 0
for every point λ ∈ C+.
The class Mℓ(jpq) of left γ-generating matrices was introduced in [8] as
the set of mvf’s A(µ) which satisfies (1), (2) and
(3ℓ) s12 := a12a
−1
22 ∈ Sp×q.
As was shown in [1,2], any solution of a completely indeterminate matrix
Nehari problem can be represented in the form
f(µ) = TA[s] = (a11(µ)s(µ) + a12(µ))(a21(µ)s(µ) + a22(µ))
−1, (1.1)
where A ∈ Mr(jpq), and s is a mvf of the Schur class Sp×q.
A mvf A ∈ Mr(jpq) is said to be right singular γ-generating matrix
if TA[Sp×q] ⊂ Sp×q. A mvf A ∈ Mr(jpq) is said to be right regular
γ-generating matrix if the factorization A = A1A2 with a factor A1 ∈
Mr(jpq) and a right singular factor A2 implies that A2 is constant. These
two subclasses of Mr(jpq) will be designated Mr,S(jpq) and Mr,R(jpq),
respectively.
Similarly, the classes Mℓ,S(jpq) and Mℓ,R(jpq) were introduced in [8,
10] and in fact the classes Mr,S(jpq) and Mℓ,S(jpq) coincide:
MS(jpq) := Mr,S(jpq) = Mℓ,S(jpq).
As was shown in [8] a resolvent matrix A which describes solutions of
the Nehari problem is a right regular γ-generating matrix.
In [8] it was shown that any γ-generating matrix admits a factoriza-
tion
A = A1A2, where A1 ∈ Mr,R(jpq), A2 ∈ MS(jpq).
Classes Mr
κ(jpq) and Mℓ
κ(jpq) of generalized γ-generating matrices
were introduced in [14,20], where also connections between generalized γ-
generating matrices of the class Mr
κ(jpq) (resp. Mℓ
κ(jpq)) and generalized
jpq-inner mvf’s of the class Ur
κ(jpq) (resp. U ℓ
κ(jpq)) were established.
Sufficient conditions for regular–singular factorization of generalized
jpq-inner mvf were found in [15]. In the present paper the notions of
singular and regular right and left generalized γ-generating mvf’s are
introduced and studied.
Sufficient conditions for existance of regular-singular factorization for
right and left generalized γ-generating mvf’s are also found.
O. Sukhorukova 577
1.1. The generalized Schur class
Let Ω+ be equal to either D = {λ ∈ C : |λ| < 1} or C+ = {λ ∈ C :
−i(λ− λ̄) > 0}. Let us set
ρω(λ) =
{
1− λω, if Ω+ = D;
−2πi(λ− ω), if Ω+ = C+.
and let Ω− := {ω ∈ C : ρω(ω) < 0}. Then Ω0 := ∂Ω+ is either the unit
circle T, if Ω+ = D, or the real line R, if Ω+ = C+.
Let κ ∈ Z+. Recall [5], that a Hermitian kernel Kω(λ) : Ω × Ω →
Cm×m is said to have κ negative squares, if for every positive integer n
and every choice of ωj ∈ Ω and uj ∈ Cm (j = 1, . . . , n) the matrix
(u∗kKωj (ωk)uj)
n
j,k=1
has at most κ, and for some choice of n ∈ N, ωj ∈ Ω and uj ∈ Cm exactly
κ negative eigenvalues.
Denote by hs the domain of holomorphy of the mvf s(λ) and let us
set h±s := hs ∩ Ω±.
Let Sq×p
κ denote the generalized Schur class of q × p mvf’s s that are
meromorphic in Ω+ and for which the kernel
Λs
ω(λ) =
Ip − s(λ)s(ω)∗
ρω(λ)
(1.2)
has κ negative squares on h+s × h+s (see [17]). In the case where κ = 0
the class Sq×p
0 coincides with the Schur class Sq×p. A mvf s ∈ Sq×p is
said to be inner (s ∈ Sq×p
in ), if Ip − s(µ)∗s(µ) = 0 for a.e. point µ ∈ Ω0.
Mvf s ∈ Sq×p is said to be outer (s ∈ Sq×p
out ), if sHp
2 = Hq
2 .
As was shown in [17] every mvf s ∈ Sq×p
κ admits a factorization of
the form
s(λ) = bℓ(λ)
−1sℓ(λ), λ ∈ h+s , (1.3)
where bℓ ∈ Sq×q
in is a q × q BP (Blaschke–Potapov) product of degree κ
(see. [10]), sℓ ∈ Sq×q and
rank
[
bℓ(λ) sℓ(λ)
]
= q (λ ∈ Ω+). (1.4)
The representation (1.3) is called a left KL (Krein–Langer) factorization.
Similarly, every generalized Schur function s ∈ Sq×p
κ admits a right KL-
factorization
s(λ) = sr(λ)br(λ)
−1 for λ ∈ h+s , (1.5)
578 Factorization of generalized γ-generating matrices
where br ∈ Sp×p is a BP-product of degree κ, sr ∈ Sq×p and
rank
[
br(λ)
∗ sr(λ)
∗
]
= p (λ ∈ Ω+). (1.6)
Recall the notations (see [10]): Rp×q – the class of rational p× q mvf’s,
N p×q
± = {f = h−1g : g ∈ Hp×q
∞ (Ω±), h ∈ S1×1
out (Ω±)};
N p×q
out = {f = h−1g : g ∈ Sp×q
out , h ∈ S1×1
out }.
The limit values f(µ) of mvf f(λ) ∈ N p×q(C+) (N p×q(D)) are defined
a.e. on R (T)
f(µ) = lim
ν↓0
f(µ+ iν) (f(µ) = lim
r↑1
f(rµ)). (1.7)
Similarly, the limit values of f ∈ N p×q(Ω−) are defined a.e. on Ω0.
Definition 1.1. A p×q mvf f− in Ω− is said to be a pseudocontinuation
of a mvf f ∈ N p×q, if
(1) f#− ∈ N p×q;
(2) f−(µ) = f(µ) a.e. on Ω0.
The subclass of all mvf’s f ∈ N p×q that admit pseudocontinuations f−
into Ω− will be denoted Πp×q. Sometimes the superindex p× q is dropped
and we denote this class by Π if it does not lead to confusion.
1.2. Generalized jpq-inner mvf’s
Definition 1.2. [4,13] An m×m mvf W (λ) that is meromorphic in Ω+
is said to belong to the class Uκ(jpq) of generalized jpq-inner mvf’s, if:
(i) the kernel
KW
ω (λ) =
jpq −W (λ)jpqW (ω)∗
ρω(λ)
(1.8)
has κ negative squares in h+W × h+W , where h+W denotes the domain
of holomorphy of W in Ω+ and
(ii) jpq −W (µ)jpqW (µ)∗ = 0 a.e. on the boundary Ω0 of Ω+.
Let us recall some facts concerning the PG (Potapov–Ginzburg) trans-
form of generalized jpq-inner mvf’s. As is known [4, Theorem 6.8], for
O. Sukhorukova 579
every W ∈ Uκ(jpq) the matrix w22(λ) is invertible for all λ ∈ h+W except
for at most κ point in Ω+. The PG-transform S = PG(W ) of W (see [3])
S(λ) :=
[
w11(λ) w12(λ)
0 Iq
][
Ip 0
w21(λ) w22(λ)
]−1
(1.9)
is well defined for those λ ∈ h+W , for which w22(λ) is invertible, S(λ)
belongs to the class Sm×m
κ and S(µ) is unitary for a.e. µ ∈ Ω0 (see [4,13]).
The formula (1.9) can be rewritten as
S =
[
s11 s12
s21 s22
]
=
[
w11 − w12w
−1
22 w21 w12w
−1
22
−w−1
22 w21 w−1
22
]
. (1.10)
Since the mvf S(λ) has unitary nontangential boundary limits a.e. on
Ω0, the pseudocontinuation of S to Ω− can be defined by the formula
S(λ) = (S#(λ))−1, where the reflection function S#(λ) is defined by
S#(λ) = S(λ◦)∗, λ◦ =
{
1/λ : if Ω+ = D, λ ̸= 0;
λ : if Ω+ = C+.
(1.11)
1.3. The class Ur
κ(jpq)
Definition 1.3. [13] An m×m mvf W (λ) ∈ Uκ(jpq) is said to be in the
class Ur
κ(jpq), if
s21 := −w−1
22 w21 ∈ Sq×p
κ . (1.12)
Theorem 1.4. [13] Let W ∈ Ur
κ(jpq) and let the BP-factors bℓ and br be
defined by the KL-factorizations of s21:
s21(λ) := bℓ(λ)
−1sℓ(λ) = sr(λ)br(λ)
−1, λ ∈ h+s21 , (1.13)
where bℓ ∈ Sq×q
in , br ∈ Sp×p
in , sℓ, sr ∈ Sq×p. Then the mvf’s bℓs22 and
s11br are holomorphic in Ω+, and hence they admit the following inner-
outer and outer-inner factorizations
s11br = b1a1, bℓs22 = a2b2, (1.14)
where b1 ∈ Sp×p
in , b2 ∈ Sq×q
in , a1 ∈ Sp×p
out , a2 ∈ Sq×q
out .
The pair {b1, b2} is called the right associated pair of the mvf W ∈
Ur
κ(jpq) and is written as {b1, b2} ∈ apr(W ). In the case κ = 0 this notion
was introduced in [6].
580 Factorization of generalized γ-generating matrices
Proposition 1.5. [13, 16] If s ∈ Sq×p, then there exists a set of mvf’s
cℓ ∈ Hq×q
∞ , dℓ ∈ Hp×q
∞ , cr ∈ Hp×p
∞ and dr ∈ Hp×q
∞ , such that[
cr dr
−sℓ bℓ
][
br −dℓ
sr cℓ
]
=
[
Ip 0
0 Iq
]
. (1.15)
If, in addition, s ∈ Π, then cℓ, dℓ, cr, dr can be chosen from Π.
Proof. The first statement was proved in [13, Theorem 4.9] (the rational
case was treated in [16]).
Assume now that s ∈ Π and hence also sℓ ∈ Π. Let dℓ be a rational
mvf’s such that
b−1
ℓ (Iq − sℓdℓ) ∈ Hq×q
∞ .
Such a mvf can be chosen via matrix Lagrange–Silvester interpolation.
Then by setting
cℓ := b−1
ℓ (Ip − sℓdℓ)
one obtains cℓ ∈ Hq×q
∞ ∩Πq×q, since bℓ, sℓ, dℓ ∈ Π.
The inclusions cr, dr ∈ Π are implied by (1.15).
By [13, Theorem 4.11] for everyW ∈ Ur
κ(jpq) and cℓ and dℓ as in (1.15)
the mvf
K = (−w11dℓ + w12cℓ)(−w21dℓ + w22cℓ)
−1, (1.16)
belongs to Hp×q
∞ and admits the representations
K = (−w11dℓ + w12cℓ)a2b2, (1.17)
where {b1, b2} ∈ apr(W ) and a2 ∈ Sq×q
out is determined by (1.14).
1.4. The class U ℓ
κ(jpq)
The following definitions and statements concerning the dual class
U ℓ
κ(jpq) are taken from [19].
Definition 1.6. An m ×m mvf W ∈ Uκ(jpq) is said to be in the class
U ℓ
κ(jpq), if
s12 := w12w
−1
22 ∈ Sp×q
κ . (1.18)
If W ∈ Uκ(jpq) and the mvf W̃ is defined by
W̃ (λ) =
{
W (λ)∗, if Ω+ = D,
W (−λ)∗ if Ω+ = C+.
(1.19)
O. Sukhorukova 581
then, as was shown in [19], the following equivalence holds:
W ∈ U ℓ
κ(jpq) ⇐⇒ W̃ ∈ Ur
κ(jpq) (1.20)
and as a corollary of Theorem 1.4 one can get the following statement.
Theorem 1.7. Let W ∈ U ℓ
κ(jpq) and let the BP-factors bℓ and br be
defined by the KL-factorizations of s12:
s12(λ) = bℓ(λ)
−1sℓ(λ) = sr(λ)br(λ)
−1, (λ ∈ h+s12), (1.21)
where bℓ ∈ Sp×p
in , br ∈ Sq×q
in , sℓ, sr ∈ Sp×q. Then
s22br ∈ Sq×q and bℓs11 ∈ Sp×p. (1.22)
Definition 1.8. Consider inner-outer and outer-inner factorizations of
bℓs11 and s22br
bℓs11 = a1b1, s22br = b2a2, (1.23)
where b1 ∈ Sp×p
in , b2 ∈ Sq×q
in , a1 ∈ Sp×p
out , a2 ∈ Sq×q
out . The pair b1, b2 of
inner factors in the factorizations (1.23) is called the left associated pair
of the mvf W ∈ U ℓ
κ(jpq) and is written as {b1, b2} ∈ apℓ(W ), for short.
Remark 1.9. As was shown in [19] (3.25) if {b1, b2} ∈ apℓ(W ), then
s̃11b̃ℓ = b̃1ã1, b̃rs̃22 = ã2b̃2, and, therefore, {b̃1, b̃2} ∈ apr(W̃ ).
As was shown in [19], there exists a set of mvf’s cℓ ∈ Hp×p
∞ , dℓ ∈ Hq×p
∞ ,
cr ∈ Hq×q
∞ and dr ∈ Hq×p
∞ , such that[
cℓ sr
dℓ br
][
bℓ −sℓ
−dr cr
]
=
[
Ip 0
0 Iq
]
. (1.24)
1.5. Reproducing kernel Pontryagin spaces
In this subsection we review some facts and notation from [11–13]
on the theory of indefinite inner product spaces for the convenience of
the reader. A linear space K equipped with a sesquilinear form ⟨·, ·⟩K on
K ×K is called an indefinite inner product space. A subspace F of K is
called positive (resp. negative) if ⟨f, f⟩K > 0, (resp. < 0) for all f ∈ F ,
f ̸= 0.
An indefinite inner product space (K, ⟨·, ·⟩K) is called a Pontryagin
space, if it can be decomposed as the orthogonal sum
K = K+ ⊕K− (1.25)
582 Factorization of generalized γ-generating matrices
of a positive subspace K+ which is a Hilbert space with respect to the
inner product ⟨·, ·⟩K and a negative subspace K− of finite dimension. The
number ind−K := dimK− is referred to as the negative index of K.
The isotropic part of L ⊂ K is defined by L0 := {x ∈ L : ⟨x, y⟩L =
0, y ∈ L}. The subspace L is called nondegenerate iff L0 = {0}.
A Pontryagin space (K, ⟨·, ·⟩K) of Cm-valued functions defined on a
subset Ω of C is called a RKPS (reproducing kernel Pontryagin space),
if there exists a Hermitian kernel Kω(λ) : Ω× Ω → Cm×m, such that:
(1) for every ω ∈ Ω and every u ∈ Cm the vvf Kω(λ)u belongs to K;
(2) for every f ∈ K, ω ∈ Ω and u ∈ Cm the following identity holds
⟨f,Kωu⟩K = u∗f(ω). (1.26)
It is known (see [18]) that for every Hermitian kernel Kω(λ) : Ω×Ω →
Cm×m with a finite number κ of negative squares on Ω × Ω there is
a unique Pontryagin space K with reproducing kernel Kω(λ), and that
ind−K = sq−K = κ. In the case κ = 0 this fact is due to Aronszajn [5].
For W ∈ Uκ(jpq) we denote by K(W ) the RKPS associated with the
kernel KW
ω (λ) defined by (1.8).
2. A-regular–A-singular factorization of generalized
J-inner mvf’s
A mvf W ∈ Uκ(jpq) is called A-singular, if it is an outer mvf (see [6,
19]). The set of A-singular mvf’s in Uκ(jpq) is denoted by US
κ (jpq).
We will be also using the following subclasses of the class US
κ (jpq):
Ur,S
κ (jpq) := Ur
κ(jpq) ∩Nm×m
out , U ℓ,S
κ (jpq) := U ℓ
κ(jpq) ∩Nm×m
out .
In the case κ = 0 the class US(jpq) := US
0 (jpq) was introduced and
characterized in terms of associated pairs by D. Arov in [9]. For κ ̸= 0 a
definition of A-singular generalized jpq-inner mvf and its characterization
in terms of associated pairs was given in [19].
Theorem 2.1. [19] Let W ∈ Ur
κ(jpq) and let {b1, b2} ∈ apr(W ). Then:
W ∈ Ur,S
κ (jpq) ⇐⇒ b1 ≡ const, b2 ≡ const.
Theorem 2.2. [19] Let W ∈ U ℓ
κ(jpq) and let {b1, b2} ∈ apℓ(W ). Then:
W ∈ U ℓ,S
κ (jpq) ⇐⇒ b1 ≡ const, b2 ≡ const.
O. Sukhorukova 583
Lemma 2.3. Let W ∈ Ur
κ(jpq) and let {b1, b2} ∈ apr(W ). Then:
W ∈ Ur,S
κ (jpq) ⇐⇒ W̃ ∈ U ℓ,S
κ (jpq).
Proof. Let W ∈ Ur,S
κ (jpq). Then by Theorem 2.1,
b1 ≡ const, b2 ≡ const.
Due to Remark 1.9 one obtains b̃1 ≡ const, b̃2 ≡ const and hence W̃ ∈
U ℓ,S
κ (jpq) by Theorem 2.2. The proof of the converse is similar.
Lemma 2.4. [15] Let a mvf W ∈ Ur
κ(jpq) admits a factorization
W =W (1)W (2), W (1) ∈ Uκ1(jpq), W (2) ∈ Uκ2(jpq), (2.1)
where κ1 + κ2 = κ. Then:
(i) W (1) ∈ Ur
κ1
(jpq);
(ii) For {b1, b2} ∈ apr(W ) and {b(1)1 , b
(1)
2 } ∈ apr(W (1)) one has
θ1 := (b
(1)
1 )−1b1 ∈ Sp×p
in , θ2 := b2(b
(1)
2 )−1 ∈ Sq×q
in . (2.2)
Definition 2.5. [15] A mvf W ∈ Ur
κ(jpq) is called right A-regular, if for
any factorization
W =W (1)W (2), W (1) ∈ Ur
κ1
(jpq), W (2) ∈ U ℓ
κ2
(jpq), (2.3)
with κ1 + κ2 = κ the assumption W (2) ∈ U ℓ,S
κ2 (jpq) implies W (2)(λ) ≡
const.
Similarly, a mvf W ∈ U ℓ
κ(jpq) is called left A-regular, if for any
factorization (2.3) with κ1 + κ2 = κ the assumption W (1) ∈ US
κ1
(jpq)
implies W (1)(λ) ≡ const.
In the case κ = 0 Definition 2.5 is simplified since Ur
0 (jpq) = U ℓ
0(jpq) =
U(jpq) (see [7]).
In the next lemma we present one necessary and one sufficient condi-
tion for a mvf W (λ) ∈ Ur
κ(jpq) to be regular. Let us set
LW := K(W ) ∩ Lm
2 . (2.4)
Lemma 2.6. [15] Let W ∈ Ur
κ(jpq), let K(W ) be the RKPS with the
kernel KW
ω (λ), defined by (1.8), let ind−LW = κ and let κ1 = ind−(LW ),
κ2 = κ− κ1. Assume that:
584 Factorization of generalized γ-generating matrices
(A1) hW ∩ Ω0 ̸= ∅;
(A2) The closure LW of LW is nondegenerate in K(W ).
Then the following implications hold:
(1) W ∈ Ur,R
κ (jpq) =⇒ LW = K(W );
(2) K(W̃ ) ⊂ Lm×m
2 =⇒W ∈ Ur,R
κ (jpq).
Denote by Rm×m the set of rational m×m-mvf’s. The following
criterion for a rational mvf W ∈ Ur
κ(jpq) to be right A-regular is given
in [15]. We will present here a simpler proof of this result.
Theorem 2.7. Let W ∈ Ur
κ(jpq) be a rational mvf. Then
(1) W ∈ Ur,R
κ (jpq) ⇐⇒ LW = K(W ).
(2) W ∈ Ur,R
κ (jpq) ⇐⇒W ∈ L̃m×m
2 .
Proof. Let W ∈ Ur,R
κ (jpq) ∩ Rm×m. Then by Lemma 2.6 LW = K(W ),
and sinceW is rational, LW = LW = K(W ). Therefore, K(W ) ⊂ Lm×m
2 .
Hence W ∈ L̃m×m
2 . The converse is immediate from Lemma 3.19(3)
in [15].
Lemma 2.8. Let W ∈ Ur
κ(jpq). Then:
W ∈ Ur,R
κ (jpq) ⇐⇒ W̃ ∈ U ℓ,R
κ (jpq).
Proof. Let W ∈ Ur
κ(jpq) and assume that W̃ = W̃ (1)W̃ (2), where W̃ (1) ∈
Ur,S
κ1 (jpq), W̃
(2) ∈ U ℓ
κ2
(jpq). Then
W =W (2)W (1), where W (1) ∈ U ℓ,S
κ1
(jpq), W (2) ∈ Ur
κ2
(jpq).
By the regularity of W , W (1) ≡ const. Hence W̃ (1) ≡ const and thus
W̃ ∈ U ℓ,R
κ (jpq). The converse implication is obtained similarly.
The following theorem was proved in [15].
Theorem 2.9. Let W ∈ Ur
κ(jpq) ∩ U ℓ
κ(jpq) ∩Rm×m. Then the following
statements are equivalent:
(1) W admits the factorization
W =W (1)W (2), W (1) ∈ Ur,R
κ1
(jpq), W (2) ∈ U ℓ,S
κ2
(jpq) (2.5)
with κ = κ1 + κ2;
O. Sukhorukova 585
(2) LW is a nondegenerate subspace of K(W ), ind−LW = κ1.
Moreover, if (2) is the case then the factors W (1) and W (2) in (2.5) are
uniquely determined up to jpq-unitary factors.
In the classical case (κ = 0) this result coincides with the factorization
Theorem in [10].
Let us present now an analog of Theorem 2.9 for A-singular-A-regular
factorizations.
Corollary 2.10. Let W ∈ Ur
κ(jpq)∩U ℓ
κ(jpq)∩Rm×m. Then the following
statements are equivalent:
(1) W admits the factorization
W =W (2)W (1), W (1) ∈ U ℓ,R
κ1
(jpq), W (2) ∈ Ur,S
κ2
(jpq) (2.6)
with κ = κ1 + κ2;
(2) L
W̃
is a nondegenerate subspace of K(W̃ ), ind−LW̃
= κ1.
Moreover, if (2) is the case then the factors W (1) and W (2) in (2.5) are
uniquely determined up to jpq-unitary factors.
Proof. Assume that (2) holds and consider the mvf W̃ ∈ Ur
κ(jpq) ∩
U ℓ
κ(jpq) ∩Rm×m see (1.20). By Theorem 2.9
W̃ = W̃ (1)W̃ (2), where W̃ (1) ∈ Ur,R
κ1
(jpq), W̃
(2) ∈ U ℓ,S
κ2
(jpq), (2.7)
with κ1 + κ2 = κ. Hence by Lemma 2.3 and 2.8 W admits the factoriza-
tion (2.6). Conversely, let (1) holds. Then by (1.20), Lemma 2.3 and 2.8
the mvf W̃ admits the factorization (2.7) and hence by Theorem 2.9 the
statement (2) holds.
The following example illustrates importance of the condition (2) of
Theorem 2.9.
Example 2.11. Let
W1(λ) =
1
2λ− 2
[
λ2 − 3λ+ 1 λ2 − λ+ 1
λ2 − λ+ 1 λ2 − 3λ+ 1
]
.
As was shown in [15], this mvf W1 belongs to Ur
1 (j11) ∩ U ℓ
1(j11) and it
does not admit the A-regular –A-singular factorization.
The RKPS K(W1) and the subspace LW1 take the form
586 Factorization of generalized γ-generating matrices
K(W1) = span
{[
1
1
]
,
1
λ− 1
[
1
−1
]}
, LW1 = span
{[
1
1
]}
.
and LW1 is a degenerate subspace of K(W1) see [15]. Therefore, condition
(2) of Theorem 2.9 does not holds. By Corollary 2.10 W1 does not admit
an A-singular–A-regular factorization.
3. Generalized γ-generating matrices
Definition 3.1. Let Mr
κ(jpq) denote the class of m×m mvf’s
A(µ) =
[
a11(µ) a12(µ)
a21(µ) a22(µ)
]
, (3.1)
with blocks a11 of size p× p and a22 of size q × q such that:
(1) A(µ) is a measurable on Ω0 mvf that is jpq-unitary a.e. on Ω0;
(2) s21 = −a−1
22 a21 ∈ Sq×p
κ ;
(3) (a#11)
−1br = a1 ∈ Sp×p
out , bℓa
−1
22 = a2 ∈ Sq×q
out , where bℓ, br are BP-
products of degree κ which are determined by KL-factorizations of
s21.
The mvf’s in the class Mr
κ(jpq) are called generalized right γ-generating
matrices.
Definition 3.2. Let Mℓ
κ(jpq) denote the class of m ×m mvf’s A(µ) of
the form (3.1), such that:
(1) A(µ) is a measurable on Ω0 mvf that is jpq-unitary a.e. on Ω0;
(2) s12 = a12a
−1
22 ∈ Sp×q
κ ;
(3) bℓ(a
#
11)
−1 = a1 ∈ Sp×p
out , a
−1
22 br = a2 ∈ Sq×q
out , where bℓ, br are BP-
product of degree κ which are determined by KL-factorizations of
s12.
The mvf’s in the class Mℓ
κ(jpq) are called generalized left γ-generating
matrices.
Definition 3.3. An ordered pair {b1, b2} of inner mvf’s b1 ∈ N p×p,
b2 ∈ N q×q is called a denominator of the mvf f ∈ N p×q, if b1fb2 ∈ N p×q
+ .
The set of denominators of f will be denoted by den(f).
O. Sukhorukova 587
Theorem 3.4. Let A ∈ Mr
κ(jpq), let bℓ, sℓ, br, sr be defined by KL-facto-
rization of s21 ∈ Sq×p
κ . Let cℓ, dℓ, cr, dr be defined by (1.15) and let
f r0 := (−a11dℓ + a12cℓ)(−a21dℓ + a22cℓ)
−1 = (−a11dℓ + a12cℓ)a2. (3.2)
Then:
(i) if den(f r0 ) ̸= ∅ and {b1, b2} ∈ den(f r0 ) then
W (z) =
[
b1 0
0 b−1
2
]
A(z) ∈ Ur
κ(jpq), {b1, b2} ∈ apr(W ) (3.3)
and hence A ∈ Πm×m. Conversely, if
W ∈ Ur
κ(jpq) and {b1, b2} ∈ apr(W ). (3.4)
then
A(z) =
[
b−1
1 0
0 b2
]
W (z) ∈ Πm×m∩Mr
κ(jpq) and {b1, b2} ∈ den(f r0 ).
(ii) if A ∈ Πm×m then den(f r0 ) ̸= ∅ and, moreover, for some choice of
mvf’s cℓ, dℓ, cr, dr in (1.15) one gets f r0 ∈ Π.
(iii) if {c(i)ℓ , d
(i)
ℓ , c
(i)
r , d
(i)
r } (i = 1, 2) are two sets of mvf’s satisfying
(1.15) and
f r,i0 = (−a11d(i)ℓ + a12c
(i)
ℓ )a2, i ∈ {1, 2} (3.5)
then den(f r,10 ) = den(f r,20 ).
Proof. (i) The first implication holds by Theorem 4.3 from [14]. The
converse implication follows from Theorem 4.3 and from the fact that
W ∈ Πm×m since W is jpq-unitary. By virtue of
[
b−1
1 0
0 b2
]
∈ Πm×m,
this implies A ∈ Πm×m.
(ii) Since A ∈ Πm×m one has a11, a12, a2 ∈ Π. By Proposition 1.5 the
mvf’s cℓ and dℓ can be chosen from Π. Therefore, f r0 ∈ Π.
(iii) Let {b1, b2} ∈ den(f r,10 ) and let W (z) be given by (3.3). Then by
item (i) W ∈ Ur
κ(jpq) and {b1, b2} ∈ apr(W ). Let us set
K(i) = (−w11d
(i)
ℓ + w12c
(i)
ℓ )a2b2, i = {1, 2}.
588 Factorization of generalized γ-generating matrices
Then by [13, Theorem 4.11]
(b1a1)
−1(K(1) −K(2))(a2b2)
(−1) ∈ Hp×q
∞ . (3.6)
Since K(i) = b1f
r,i
0 b2 (i = 1, 2) one gets from (3.6)
f r,10 − f r,20 ∈ Hp×q
∞ .
Therefore, {b1, b2} ∈ den(f r,20 ). Clearly, the converse implication is also
true.
Remark 3.5. A similar assertion also holds for the class of generalized
left γ-generating matrices. Let A ∈ Mℓ
κ(jpq), bℓ, sℓ, br, sr be defined by
KL-factorization of s12 ∈ Sq×p
κ . Let cℓ, dℓ, cr, dr defined by (1.24) and let
f ℓ0 := a2(−dra11 + cra21) = (−dra12 + cra22)
−1(−dra11 + cra21). (3.7)
Then:
(i) if den(f ℓ0) ̸= ∅ and {b1, b2} ∈ den(f ℓ0) then
W (z) = A(z)
[
b1 0
0 b−1
2
]
∈ U ℓ
κ(jpq) (3.8)
and {b1, b2} ∈ apℓ(W ). Conversely, if
W ∈ U ℓ
κ(jpq) and {b1, b2} ∈ apℓ(W ), (3.9)
then
A(z) =W (z)
[
b−1
1 0
0 b2
]
∈ Πm×m∩Mℓ
κ(jpq) and {b1, b2} ∈ den(f ℓ0).
(ii) if A ∈ Πm×m then denf ℓ0 ̸= ∅ and, moreover, the mvf’s cℓ, dℓ, cr, dr
in (1.24) can be chosen from Π and then f ℓ0 ∈ Π.
(iii) {c(i)ℓ , d
(i)
ℓ , c
(i)
r , d
(i)
r } (i = 1, 2) two sets of mvf’s defined by (1.24)
f ℓ,i0 = a2(−dra11 + cra21), {b1, b2}α2(−d(i)r a11 + c(i)r a21),
{b1, b2} ∈ denf ℓ,i0 , i = {1, 2},
then denf ℓ,10 = denf ℓ,20 .
O. Sukhorukova 589
Definition 3.6. We define the denominator of generalized right γ-gene-
rating mvf A ∈ Πm×m ∩Mr
κ(jpq) as
denr(A) := denf r0 ,
and the denominator of left generalized γ-generating mvf A ∈ Πm×m ∩
Mℓ
κ(jpq) as
denℓ(A) := denf ℓ0.
Definition 3.7. Let a mvf A ∈ Mr
κ(jpq) is said to be
(1) right singular and is written as A ∈ Mr,S
κ if f r0 = (−a11dℓ +
a12cℓ)a2 ∈ Hp×q
∞ ,
(2) right regular and is written as A ∈ Mr,R
κ if the factorization A =
A1A2, with A1 ∈ Mr
κ1
(jpq) and A2 ∈ Mℓ,S
κ2 (jpq), κ1+κ2 = κ implies
that A2 ≡ const.
Definition 3.8. Let a mvf A ∈ Mℓ
κ(jpq) is said to be
(1) left singular and is written as A ∈ Mℓ,S
κ if f ℓ0 = a2(−dra11+cra21) ∈
Hp×q
∞ ,
(2) left regular and is written as A ∈ Mℓ,R
κ if the factorization A =
A2A1, with A1 ∈ Mℓ
κ1
(jpq) and A2 ∈ MrS
κ2
(jpq), κ1+κ2 = κ implies
that A2 ≡ const.
In the case κ = 0, the left singularity coincides with the right singu-
larity, therefore our definition coincides with the definition in [8].
4. Fatorization of γ-generating matrices
Lemma 4.1. Let A ∈ Mr
κ(jpq) ∩Πm×m. Then:
A ∈ Mr,S
κ (jpq) ⇐⇒ A ∈ Ur,S
κ (jpq).
Proof. Let A ∈ Mr,S
κ (jpq), then f0 = (−a11dℓ + a12cℓ)a2 ∈ Hp×p
∞ , there-
fore {Ip, Iq} ∈ denf r0 . In view of Theorem 3.4 this implies A ∈ Ur
κ(jpq)
and {Ip, Iq} ∈ apr(A). Hence by Theorem 2.1 A ∈ Ur,S
κ (jpq).
Conversely, if A ∈ Ur,S
κ (jpq) and {b1, b2} ∈ apr(A), then bi ≡ const,
i = {1, 2}. Hence by Theorem 3.4 A ∈ Mr
κ(jpq), {b1, b2} ∈ denf r0 , i.e.
f r0 ∈ Hp×q
∞ , and thus A ∈ Mr,S
κ (jpq)
Corollary 4.2. Let A ∈ Mℓ
κ(jpq) ∩Πm×m. Then:
A ∈ Mℓ,S
κ (jpq) ⇐⇒ A ∈ U ℓ,S
κ (jpq).
590 Factorization of generalized γ-generating matrices
Proof. Let A ∈ Mℓ,S
κ (jpq), then à ∈ Mr,S
κ (jpq), hence by Lemma 4.1
à ∈ Ur,S
κ (jpq), and thus A ∈ U ℓ,S
κ (jpq). Analogously, the assumption
A ∈ U ℓ,S
κ implies A ∈ Mℓ,S
κ .
Lemma 4.3. Let A′,A ∈ Mr
κ(jpq) and A′ =
[
θ−1
1 0
0 θ2
]
A, θ1 ∈ Sp×p
in ,
θ2 ∈ Sq×q
in . Then θ1 ≡ const, θ2 ≡ const.
Proof. Let A′(µ) =
[
a′11(µ) a′12(µ)
a′21(µ) a′22(µ)
]
and let the mvf A(µ) has block
representation (3.1). Then
A′ =
[
θ−1
1 0
0 θ2
]
A =
[
θ−1
1 a11 θ−1
1 a12
θ2a21 θ2a22
]
,
and hence by Definition 3.1
s′21 := −(a′22)
−1a′21 = −a−1
22 θ
−1
2 θ2a21 = −a−1
22 a21 = s21 ∈ Sq×p
κ .
This means that the Krein-Langer factorizations of s21 and s′21 coincide
s′21 = s21 = b−1
ℓ sℓ = srb
−1
r ,
where bℓ ∈ Sq×q
in , br ∈ Sq×q
in , sℓ, sr ∈ Sq×p. Hence
a′1 = (a′11)
−#br = θ#1 (a11)
−#br ∈ Sp×p
out , and a1 = a−#
11 br ∈ Sq×p
out .
This is possible only when θ1 ≡ const. Analogously,
a′2 = bℓ(a
′
22)
−1 = bℓa
−1
22 θ
−1
2 ∈ Sq×q
out and a2 = bℓa
−1
22 ∈ Sq×q
out
consequently θ2 ≡ const.
Lemma 4.4. Let a mvf A ∈ Mr
κ(jpq) ∩Πm×m admits the factorization
A = A(1)A(2), where A(1) ∈ Mr
κ1
(jpq), A(2) ∈ Mℓ,S
κ2
(jpq), (4.1)
with κ1 + κ2 = κ. Then denr(A(1)) ⊂ denr(A).
Proof. Let a mvf A ∈ Mr
κ(jpq) ∩ Πm×m admits the factorization (4.1).
Since A(2) ∈ Mℓ,S
κ2 , then f0 ∈ H∞ and then A(2) ∈ Πm×m. Therefore
A(1) = A(A(2))−1 and thus A(1) ∈ Π. Let {b(1)1 , b
(1)
2 } ∈ denr(A(1)) and
κ1 + κ2 = κ. By Theorem 3.4
W (1) =
[
b
(1)
1 0
0 (b
(1)
2 )−1
]
A(1) ∈ Ur
κ1
(jpq), {b(1)1 , b
(2)
2 } ∈ aprW (1),
O. Sukhorukova 591
W (2) = A(2) ∈ U ℓ,S
κ2
(jpq).
Let us set
W ′ :=
[
b
(1)
1 0
0 (b
(1)
2 )−1
]
A =
[
b
(1)
1 0
0 (b
(1)
2 )−1
]
A(1)A(2) =W (1)W (2).
Then W ′ ∈ Uκ′ , κ′ ≤ κ1 + κ2 = κ (see [4] or [13]).
On the other hand, s21 = −(w′
22)
−1w′
21 = −a−1
22 a21 ∈ Sp×q
κ , hence
κ′ ≥ κ and therefore κ′ = κ. Then W ′ ∈ Ur
κ(jpq).
Let {b′1, b′2} ∈ apr(W ′), hence, in view of Lemma 2.4 b′1 = b
(1)
1 θ1,
b′2 = θ2b
(1)
2 . By Theorem 3.4
A′ =
[
b′−1
1 0
0 b′2
]
W ′ =
[
b′−1
1 0
0 b′2
]
W (1)W (2)
=
[
b′−1
1 0
0 b′2
][
b
(1)
1 0
0 (b
(2)
2 )−1
]
A(1)A(2) =
[
θ−1
1 0
0 θ2
]
A ∈ Mr
κ(jpq),
Hence, by Lemma 4.3 θ1 ≡ const, θ2 ≡ const. Consequently {b(1)1 , b
(1)
2 } ∈
apr(W ′). Thus by Theorem 3.4 {b(1)1 , b
(1)
2 } ∈ denr(A).
Lemma 4.5. Let a mvf A ∈ Mr
κ(jpq) ∩ L̃p×q
2 ∩ Rm×m. Then A ∈
Mr,R
κ (jpq).
Proof. Let A = A(1)A(2), where A(1) ∈ Mr
κ1
(jpq), A(2) ∈ Mℓ,S
κ2 (jpq),
κ1 + κ2 = κ. Let {b(1)1 , b
(1)
2 } ∈ denr(A(1)), then by Lemma 4.4 the pair
{b(1)1 , b
(2)
2 } ∈ denr(A). By Theorem 3.4
W (1) =
[
b
(1)
1 0
0 (b
(1)
2 )−1
]
A(1) ∈ Ur
κ1
(jpq) W (2) = A(2) ∈ U ℓ,S
κ2
(jpq),
and
W =
[
b
(1)
1 0
0 (b
(1)
2 )−1
]
A =
[
b
(1)
1 0
0 (b
(1)
2 )−1
]
A(1)A(2)
=W (1)W (2) ∈ Uκ(jpq).
Since W ∈ Uκ(jpq) ∩ L̃m×m
2 , then by Theorem 2.7(2) W ∈ Ur,R
κ (jpq).
By this condition A(2) = W (2) ≡ const. This implies A ∈ Mr,R
κ (jpq).
592 Factorization of generalized γ-generating matrices
An analogous statement for the left class Mℓ
κ(jpq) can be easily ob-
tained with the help of the transformation (1.19).
Lemma 4.6. Let a mvf A ∈ Mℓ
κ(jpq) ∩ L̃p×q
2 ∩ Rm×m. Then A ∈
Mℓ,R
κ (jpq).
Theorem 4.7. Let a mvf A ∈ Mr
κ(jpq) ∩Rm×m, {b1, b2} ∈ denr(A), let
W be given by (3.3) and let:
(1) W (z) ∈ U ℓ
κ(jpq),
(2) LW be a nondegenerate subspace of K(W ).
Then the mvf A admits regular–singular factorization
A = A(1)A(2), A(1) ∈ Mr,R
κ1
(jpq), A(2) ∈ Mℓ,S
κ2
(jpq), (4.2)
where κ1 = ind−LW and κ2 = κ− κ1.
Proof. Let condition (1) holds and let {b1, b2} ∈ ap(W ). Then by The-
orem 2.9, W admits the factorization W = W (1)W (2), where W (1) ∈
Ur,R
κ1 (jpq) and W (2) ∈ U ℓ,S
κ2 (jpq), κ = κ1 + κ2. By Theorem 3.4 W ∈
Ur
κ(jpq) and {b1, b2} ∈ apr(W ). By Theorem 2.7 (2) and hence by Lemma
3.12 [15] apr(W ) = apr(W (1)).
Hence, upon applying
[
b−1
1 0
0 b2
]
to the mvfW and by Theorem 3.4
and Lemma 4.1 we obtain
A = A(1)A(2), where A(1) ∈ Mr
κ1
(jpq), A
(2) ∈ Mℓ,S
κ2
(jpq), κ1 + κ2 = κ.
Since W (1) ∈ L̃m×m
2 , then A(1) ∈ L̃m×m
2 , and thus by Lemma 4.5 A(1) ∈
Mr,R
κ (jpq).
Theorem 4.8. Let a mvf A ∈ Mℓ
κ(jpq) ∩ R, {b1, b2} ∈ denℓ(A), let W
be given by (3.8) and let:
(1) W (z) ∈ Ur
κ(jpq),
(2) L
W̃
be a nondegenerate of K(W̃ ), with negative index ind−LW̃
=
κ1.
Then A admits regular–singular factirization
A = A(2)A(1), A(1) ∈ Mℓ,R
κ1
(jpq), A(2) ∈ Mr,S
κ2
(jpq), (4.3)
where κ1 = ind−LW̃
and κ2 = κ− κ1.
O. Sukhorukova 593
Proof. Let A ∈ Mℓ
κ(jpq), then à ∈ Mr
κ(jpq) and the mvf’s W̃ (z) and
Ã(z) satisfy the assumptions of Theorem 4.7. By Theorem 4.7 Ã admits
a factorization
à = Ã(1)Ã(2), Ã(1) ∈ Mr,R
κ1
(jpq), Ã(2) ∈ Mℓ,S
κ2
(jpq), (4.4)
where κ1 + κ2 = κ.
Using the transformation (1.19) again, we obtain (4.3).
References
[1] V. M. Adamjan, D. Z. Arov, M. G. Krein, Infinite Hankel matrices and generalized
problems of Carathodory-Fejer and I. Schur problems // Funktsional. Anal. i
Prilozhen, 2 (1968), No. 4, 1–17.
[2] V. M. Adamjan, Nondegenerate unitary couplings of semiunitary operators //
Funktsional. Anal. i Prilozhen., 7 (1973), No. 4, 1–16.
[3] D. Alpay, A. Dijksma, J. Rovnyak, H. S. V. de Snoo, Schur functions, operator
colligations, and reproducing kernel Pontryagin spaces // Oper. Theory: Adv.
Appl., 96, (1997), 229.
[4] D. Alpay, H. Dym, On applications of reproducing kernel spaces to the Schur al-
gorithm and rational J unitary factorization. I. Schur methods in operator theory
and signal processing // Oper. Theory Adv. Appl., 18 (1986), 89–159.
[5] N. Aronszajn, Theory of reproducing kernels // Trans. Amer. Math. Soc., 68
(1950), 337–404.
[6] D.Z. Arov, Realization of matrix-valued functions according to Darlington // Izv.
Akad. Nauk SSSR Ser. Mat., 37 (1973), 1299–1331.
[7] D. Z. Arov, Regular and singular J-inner matrix functions and corresponding
extrapolation problems // Funktsional. Anal. i Prilozhen., 22 (1988), No. 1, 57–
59 [in Russian]; transl. in Funct. Anal. Appl., 22 (1988), No. 1, 46–48.
[8] D. Z. Arov,γ-generating matrices, j-inner matrix-functions and related extrapo-
lation problems // Teor. Funktsii Funktsional. Anal. i Prilozhen, I, 51 (1989),
61–67; II, 52 (1989), 103–109 [in Russian]; transl. in J. Soviet Math. I, 52 (1990),
3487–3491; II, 52 (1990), 3421–3425.
[9] D. Z. Arov, H. Dym, J-inner matrix function, interpolation and inverse problems
for canonical system, I: foundation, Integral Equations // Operator Theory, 28
(1997), 1–16.
[10] D. Z. Arov, H. Dym,J-Contractive Matrix Valued Functions and Related Topics,
Cambridge University Press, Cambridge, 2008.
[11] T. Ya. Azizov, I. S. Iokhvidov, Foundations of the theory of linear operators in
spaces with an indefinite metric, Nauka, Moscow, 1986.
[12] J. Bognar, Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer
Grenzgebiete, Springer-Verlag, New York-Heidelberg, 78, 1974.
[13] V. A. Derkach, H. Dym, On linear fractional transformations associated with
generalized J-inner matrix functions // Integ. Eq. Oper. Th., 65 (2009), 1–50.
[14] V. A. Derkach, O. Sukhorukova, Generalized γ-generating matrices and Nehari-
Takagi problem // Oper. Matrices, 10 (2016), No. 4, 1073–1091.
594 Factorization of generalized γ-generating matrices
[15] V. Derkach, O. Sukhorukova, A-regular–A-singular factorizations of generalized
J-inner matrix functions //Journal of Mathematical Sciences, 23 (2017), No. 3,
231–251.
[16] B. Francis, A Cource in H∞ Control Theory, Lecture Notes in Control and In-
formation Sciences, Springer-Verlag, Berlin, 88, 1987.
[17] M. G. Krein, H. Langer, Über die verallgemeinerten Resolventen und die char-
acteristische Function eines isometrischen Operators im Raume Πκ // Hilbert
space Operators and Operator Algebras (Proc. Intern. Conf.,Tihany, 1970); Col-
loq. Math. Soc. Janos Bolyai, 5, (1972), 353–399.
[18] L. Schwartz, Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux
associes // J. Analyse Math., 13 (1964), 115–256.
[19] O. Sukhorukova, Factorization formulas for some classes of generalized J-inner
matrix valued functions // Methods Funct. Anal. Topology, 20 (2014), No. 4,
365–378.
[20] O. Sukhorukova, Generalized γ-generating matrices // Ukr. Mat. Visn, 12 (2015),
No.4, 539–558 [in Russian]; transl. in J. of Math. Sci., 218, (2016) No. 1, 89–104.
Contact information
Olena Sukhorukova National Pedagogical Dragomanov
University, Kiev, Ukraine
E-Mail: sukhorukova.elena.ua@gmail.com
|