The degenerate Carathéodory problem and the elementary multiple factor
The degenerate matricial interpolation Caratheodory problem is solved. To solve this problem we use the V.P. Potapov's approach based on the theory of J-expansive matrix-functions. The K-type subspace technique also plays an important role in these investigations.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2005
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| Цитувати: | The degenerate Carathéodory problem and the elementary multiple factor / N.N. Chernovol // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 2. — С. 225-244. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860246038545170432 |
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| author | Chernovol, N.N. |
| author_facet | Chernovol, N.N. |
| citation_txt | The degenerate Carathéodory problem and the elementary multiple factor / N.N. Chernovol // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 2. — С. 225-244. — Бібліогр.: 14 назв. — англ. |
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| container_title | Журнал математической физики, анализа, геометрии |
| description | The degenerate matricial interpolation Caratheodory problem is solved. To solve this problem we use the V.P. Potapov's approach based on the theory of J-expansive matrix-functions. The K-type subspace technique also plays an important role in these investigations.
|
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Journal of Mathematical Physics, Analysis, Geometry
2005, v. 1, No. 2, p. 225�244
The degenerate Carath�eodory problem
and the elementary multiple factor
N.N. Chernovol
Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University
4 Svobody Sq., Kharkov, 61077, Ukraine
E-mail:Nataliya.N.Chernovol@univer.kharkov.ua
Received January 26, 2005
The degenerate matricial interpolation Carath�eodory problem is solved.
To solve this problem we use the V.P. Potapov's approach based on the
theory of J-expansive matrix-functions. The K-type subspace technique
also plays an important role in these investigations.
Mathematics Subject Classi�cation 2000: 30E05, 47A56, 30D50.
Key words: matrix-valued function with nonnegative real part, interpolation Cara-
th�eodory problem, J-expansive matrix-valued function.
1. The setting of the problem. The basic matricial inequalities
connected with the problem
Let D = f� 2 C : j � j< 1g be the opened unit disk of the complex plane,
q 2 N, and let C q�q be the set of square matrices of order q with complex entries.
Denote by Cq the set of matrix-valued functions F(�) analytical in D with values
in C
q�q and satisfying the inequality
ReF(�) =
1
2
(F(�) + F�(�)) � 0
for all � 2 D . The Carath�eodory problem generalized to the matrix case (see, e.g.,
[1, 2]) is formulated in the following way.
Assume that c0; c1; : : : ; cn 2 C
q�q . Problem:
a) to �nd necessary and su�cient conditions of existence of a matrix-valued
function F(�) 2 Cq such that c0; c1; : : : ; cn are the �rst coe�cients of its
Maclaurin series:
F(�) = c0 + c1� + : : : + cn�
n + : : : ; (1:1)
c
N.N. Chernovol, 2005
N.N. Chernovol
b) to describe all function F(�) 2 Cq of the form (1.1).
The Carath�eodory problem in the scalar case (i.e., in the case q = 1) was
investigated in the papers [3, 4].
Set An = Cn + C�n, where
Cn =
2664
c0 0 : : : 0
c1 c0 : : : 0
� � � � � � � � � � � �
cn cn�1 : : : c0
3775 : (1:2)
The matrix An plays an important role when we study this problem. Namely, the
following theorems are true (see, e.g., [2, 5, 6]).
Theorem 1.1. Let fckg
1
k=0 2 C
q�q and a function F(�) is of the form (1.1).
Then F(�) 2 Cq if and only if
An =
2664
c0 + c�0 c�1 : : : c�
n
c1 c0 + c�0 : : : c�
n�1
� � � � � � � � � � � �
cn cn�1 : : : c0 + c�0
3775 � 0 (1:3)
for all n 2 N [ f0g.
This theorem is supplemented by the following
Theorem 1.2. Let An � 0 for fckg
n
k=0 2 C
q�q . Then there exists F(�) 2 Cq
such that its expansion in the Maclaurin series has the form (1.1).
Theorems 1.1, 1.2 give us the answer to the �rst question of the Carath�eodory
problem. To give an answer to the second question V.P. Potapov proposed a
special approach (see, e.g., [1, 2, 5, 7, 8]). According to this approach the basic
matricial inequality (BMI) and the dual one are corresponded to each interpo-
lation problem. The solution of each of these inequalities gives us a description
of all solutions of the problem. The BMI and the dual one for the matricial
Carath�eodory problem have the form (1.4) and (1:40) respectively.
Theorem 1.3. ([1, 2]) Let fckg
n
k=0 2 C
q�q and let F(�) be a matrix-valued
function analytical in the unite disk D . Then F(�) 2 Cq and can be represented
in the form (1.1) if and only if for the function F(�) the inequality266666664
c0 + c�0 c�1 : : : c�n F�(�) + c0
c1 c0 + c�0 : : : c�
n�1 ��[F�(�) + c0 +
c1
��
]
� � � � � � � � � � � � � � � � � � � � � � � �
cn cn�1 : : : c0 + c�0 ��n+1[F�(�) + c0 +
c1
��
+ : : : + cn
��n
]
� F(�)+F�(�)
1�� ��
377777775
� 0 (1:4)
226 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
holds everywhere in D or the following inequality266666664
c0 + c�0 c�1 : : : c�
n
1
�
[F(�)� c0]
c1 c0 + c�0 : : : c�
n�1
1
�2
[F(�)� c0 � c1�]
� � � � � � � � � � � � � � � � � � � � � � � �
cn cn�1 : : : c0 + c�0
1
�n+1
[F(�)� c0 � c1� � : : : � cn�
n]
� F(�)+F�(�)
1�� ��
377777775
� 0
(1:40)
holds everywhere in D .
Here and further the block denoted by � in the inequations of the form (1.4),
(1:40) is with the block which is adjoint to the upper right block.
Note that the matrix An of the form (1.3) is the upper left block in inequali-
ties (1.4) and (1:40). It turns out that the method of solving of these inequalities
depends on the property of the matrix An to be degenerate or not. In the case of
the nondegenerate matrix An the corresponding Carath�eodory problem is called
nondegenerate, otherwise it is called degenerate. First the nondegenerate matri-
cial Carath�eodory problem was solved constructively (i.e., directly in the terms
of the interpolation data) in [1]. In this paper V.P. Potapov's approach of solving
of the matricial interpolation problems was used (see, e.g., [7�11]. It is based on
the theory of analytical J -expansive matrix-valued functions. Note that the non-
degenerate matricial Carath�eodory problem is solved in [6] in the di�erent way.
First the constructive method of solving of the degenerate matricial interpolation
problems was obtained by investigating of the Schur problem [12]. The methods
of this paper play an important role in the Sect. 3. This section is main in the
present paper. There a constructive method of solving of the degenerate matricial
interpolation Carath�eodory problem is obtained. The main results of the paper
are formulated in Theorems 3.1 and 3.2 of this section.
In V.P. Potapov's approach the elementary multiple factor corresponding to
the BMI and the dual one plays a very important role. The parametrization of
the elementary multiple factor of the full rank connected with the nondegenerate
Carath�eodory problem is given in [1]. This parametrization is directly connected
with the parametrization of the elementary multiple factor of the full rank corre-
sponding to the nondegenerate Schur problem (see [13]). The parametrization of
an arbitrary elementary multiple factor corresponding to the Carath�eodory prob-
lem is obtained in the diploma work of L.V. Mihailova �The parametrization of
the elementary multiple factor of the nonfull rank in the Carath�eodory problem�
(Kharkov National University, 1981). There the results of the paper [14] were
used substantially. The proof of this parametrization (see the proof of Theo-
rem 2.2) is given in the present paper to deal with the main results formulated in
Theorems 3.1 and 3.2.
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 227
N.N. Chernovol
2. The parametrization of an arbitrary elementary multiple
factor
Let j be a constant hermitian involutive matrix of order N , i.e., j � = j,
j2 = I.
De�nition 2.1. (See, e.g., [1, 9]) Let B(�) be an analytical matrix-valued
function of the order N . And let it have a single pole of an arbitrary multiplicity
on the extended complex plane. B(�) is called a j-elementary multiple factor if it
is j-expansive in the unit disk and j-unitary on its boundary, i.e.,
B(�)jB�(�)� j � 0; j � j< 1 (2:1)
and
B(�)jB�(�)� j = 0; j � j= 1 (2:2)
or, equivalently,
B�(�)jB(�)� j � 0; j � j< 1
and
B�(�)jB(�)� j = 0; j � j= 1:
The Carath�eodory problem is connected with the matrix j of the form:
j = J =
�
0 Iq
Iq 0
�
: (2:3)
It can be explained, e.g., by the fact, that the matrix block standing in the lower
right angle of the left part of inequalities (1.4) and (1:40) can be presented in the
following form:
F(�) + F�(�)
1� � ��
=
1
1� � ��
[F(�); Iq]
�
0 Iq
Iq 0
� �
F�(�)
Iq
�
:
Consider a J-elementary multiple factor B(�) of the order N = 2q. Assume
that B(�) has the single pole at the point � = 0 and
B(�) = d0 +
d1
�
+ : : : +
dn+1
�n+1
; di 2 C
2q�2q ; i = 0; 1; : : : ; n+ 1: (2:4)
From conditions (2.1), (2.2) we conclude that B(�) is determined up to a J-unitary
multiplier u (uJu� = J). Further we shall assume that the following normalization
condition is ful�lled: B(1) = I.
It follows from conditions (2.1), (2.2) that rank dn+1 � q (see, e.g., [10]). If
rankdn+1 = q then B(�) is called the J-elementary multiple factor of the full
rank.
228 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
Theorem 2.1. (On a parametrization [1]) Let B(�) be a matrix-valued func-
tion of the form (2.4). It is a J-elementary multiple factor of the full rank if and
only if
B(�) = I +
1� �
�
J
�
�q;n(1) 0
0 �q;n(1)
�
H
"
��
q;n
(1�� ) 0
0 ��
q;n
(1�� )
#
;
where
�q;n(�) = [Iq; �Iq; : : : ; �
nIq]; (2:5)
H =
�
C�
I
�
(C + C�)�1 [C; I] : (2:6)
Here C is the matrix satisfying the conditions
C + C� > 0; (2:7)
CVq;n = Vq;nC (2:8)
and Vq;n is the square matrix of the (n+ 1)q-th order having the form
Vq;n =
2666664
0 : : : : : : : : : 0
Iq 0 : : : : : : 0
0 Iq 0 : : : 0
...
...
. . .
. . .
...
0 0 : : : Iq 0
3777775 : (2:9)
Moreover, the matrix C is de�ned by B(�) uniquely.
R e m a r k 2.1. One can easily see that (2.8) holds if and only if the matrix
C is a lower-triangle matrix of the form (1.2). This fact determines the connec-
tion of the J-elementary multiple factor of the full rank with the interpolation
Carath�eodory problem.
Let Q be an arbitrary hermitian matrix of the p-th order. Consider the canon-
ical basis ek = (Æik)
p
i=1, k = 1; 2; : : : ; p in C
p , where Æik is the Kronecker symbol,
i.e., Æik =
�
1; if i = k
0; if i 6= k
: We identify the matrix Q and the operator in C
p
de�ned on the basis (ek)
p
k=1 by this matrix (we denote it also by Q). Let �Q and
KerQ are the range and the kernel of the operator Q respectively. It is well-known
(see, e.g., [5]) that the operator (matrix) of the form
Q[�1]f =
�
(Q j�Q)
�1f; if f 2 �Q
0; if f 2 KerQ
(2:10)
is called the Moore�Penrose inversion of the operator (matrix) Q.
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 229
N.N. Chernovol
Theorem 2.2. 1) Let B(�) be a J-elementary multiple factor of the form
(2.4). There exist unique matrices P , C 2 C
q�q such that
P � = P; P 2 = P; (2:11)
V �
q;nP = PV �
q;nP; (2:12)
CVq;n = PVq;nC; (2:13)
C = PC; (2:14)
PC� + CP � 0; rank (PC� + CP ) = rankP: (2:15)
Moreover, B(�) can be represented in the following form:
B(�) = I +
1� �
�
J
�
�q;n(1) 0
0 �q;n(1)
�
H
"
��q;n(
1
��
) 0
0 ��q;n(
1
��
)
#
; (2:16)
where
H =
�
C�
P
�
(PC� + CP )[�1] [C;P ] ; (2:17)
�q;n(�) and Vq;n are matrices of the form (2.5) and (2.9) respectively.
2) Let conditions (2.11)�(2.15) be satis�ed and let B(�) be a matrix-valued
function of the form (2.16). Then B(�) is a J-elementary multiple factor. More-
over,
B�(�)JB(�)� J
=
1� j � j2
j � j2
"
�q;n(
1
��
) 0
0 �q;n(
1
��
)
#
H
"
��
q;n
(1�� ) 0
0 ��q;n(
1
��
)
#
: (2:18)
R e m a r k 2.2. Conditions (2.11) mean that P is an orthoprojector. In
addition, (2.12) implies that orthoprojectors projecting in C
(n+1)q on invariant
with respect to operator V �
q;n subspaces are admissible. From (2.15) we conclude
that the case of the full rank (see Theorem 2.1) is characterized by the condition
P = I(n+1)q. In fact equalities (2.11), (2.12) and (2.14) are automatically ful�lled
and relations (2.13), (2.15) transfer into relations (2.7), (2.8) in this case.
P r o o f o f T h e o r e m 2.2. Let B(�) be a J-elementary multiple factor
of the form (2.4) satisfying the normalization condition B(1) = I. This condition
allows us to represent B(�) in the form
B(�) = I +
1� �
�
�2q;n(1)D��2q;n(
1
��
);
230 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
where
D =
2664
d1 d2 : : : dn+1
d2 d3 : : : 0
: : : : : : : : : : : :
dn+1 0 : : : 0
3775 ;
and �2q;n(�) has the form analogical to (2.5). As we know ([11]), B(�) satis�es
the BMI of splitting o�264 D eJD� 1
�
D��2q;n(
1
��
)
� B�(�)JB(�)�J
1�j�j2
375 � 0; � 2 D ; (2:19)
and also the dual inequality264 D� eJD 1
��
D���2q;n(
1
�
)
� B(�)JB�(�)�J
1�j�j2
375 � 0; � 2 D nf0g; (2:20)
where
eJ =
26664
J 0 : : : 0
0 J : : : 0
...
. . .
...
0 0 : : : J
37775 ;
and J has the form (2:3). Solving inequalities (2.19) or (2.20), we obtain the
factor coinciding with B(�).
Due to [13, 14] let us simplify these inequalities before to solve them. Introduce
unitary matrix S of the form
S =
266666666664
Iq 0 0 : : : 0 0 0 : : : 0
0 0 0 : : : 0 Iq 0 : : : 0
0 Iq 0 : : : 0 0 0 : : : 0
0 0 0 : : : 0 0 Iq : : : 0
0 0 Iq : : : 0 0 0 : : : 0
: : : : : : : : : : : : : : : : : : : : : : : : : : :
0 0 0 : : : Iq 0 0 : : : 0
0 0 0 : : : 0 0 0 : : : Iq
377777777775
;
such that
J1 = S� eJS =
�
0 I(n+1)q
I(n+1)q 0
�
: (2:21)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 231
N.N. Chernovol
Let � = S�DS. Then �J1�
� = S�D eJD�S. Taking into account inequal-
ity (2.19), we have �J1�
� � 0. Denote �J1�
� =
�
a b
b� c
�
, where a; b; c 2
C
(n+1)q�(n+1)q . By analogy with [13, p. 214] we obtain the following repre-
sentation:
�J1�
� =
�
I(n+1)q X�
0
0 I(n+1)q
� �
0 0
0 eC
� �
I(n+1)q 0
X0 I(n+1)q
�
; (2:22)
where X0 is a solution of the equation cX0 = b�: Then
rankD = rank� = rank eC; eC � 0: (2:23)
In accordance to decomposition (2.21) of matrix J1 let us decompose matrix �
into blocks
� =
�
X Y
Z W
�
:
Then from (2.22) we obtain
WZ� + ZW � = eC: (2:24)
This equality implies
rankWZ� = rankZW � = rankZ = rankW = rank eC: (2:25)
It follows from (2.24) and (2.25) that
�W = �Z = �
eC
: (2:26)
Let PL be the orthoprojector in C
(n+1)q�(n+1)q onto a subspace L. Consider
the nondegenerate operator W0 = W j�W�
: �W � ! �W and a nondegenerate
transformation W1 : KerW ! KerW �. Put
Q =W�1
0 P�W +W�1
1 PKerW � :
Obviously,
Q�1 =W0P�W�
+W1PKerW :
Now de�ne the matrices C and P required in the conditions of Theorem 2.2 in
the following way:
C = QZ; P = P�W�
:
Then equalities (2.11) are ful�lled because P is a orthoprojector.
232 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
In contrast to the Schur problem (see [14]) the matrix C corresponding the
Carath�eodory problem satis�es the following condition
PC� + CP = PZ�Q� +QZP = Q(Q�1PZ� + ZPQ��1)Q�
= Q(WZ� + ZW �)Q� = Q eCQ�:
Taking into account (2.23) and (2.26), we get relations (2.15) of Theorem 2.2.
Condition (2.14) immediately follows from the de�nition of matrix C. Proper-
ties (2.12), (2.13) are established as in [14, p. 61].
To prove necessity of the conditions of Theorem 2.2 it remains to obtain (2.16).
Let
R =
�
I(n+1)q �Y Q
0 Q
�
:
According to [14], we obtain that splitting o� inequality (2.19) is equivalent to
the inequality 264 RS�D eJD�SR� 1
�
RS�D��2q;n(
1
��
)
� B�(�)JB(�)�J
1�j�j2
375 � 0: (2:27)
We have R� =
�
0 0
C P
�
. It follows from here that
RS�D eJD�SR� = R�J1�
�R� =
�
0 0
0 PC� + CP
�
;
RS�D��2q;n(
1
��
) =
�
0 0
C P
�"
��
q;n
(1�� ) 0
0 ��q;n(
1
��
)
#
:
Then (2.27) can be rewritten in the form26664
�
0 0
0 PC� + CP
�
1
�
�
0 0
C P
�"
��q;n(
1
��
) 0
0 ��q;n(
1
��
)
#
� B�(�)JB(�)�J
1�j�j2
37775 � 0: (2:28)
Put
X =
�
0 0
(PC� + CP )[�1]C (PC� + CP )[�1]P
�
:
It is evident that X is a solution of�
0 0
0 PC� + CP
�
X =
�
0 0
C P
�
:
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 233
N.N. Chernovol
Now, using V.P. Potapov's approach, we solve by the standard method inequal-
ity (2.28) (see, e.g., [13]) and obtain representation (2.16). The necessity is proved.
Let B(�) is of the form (2.16) and conditions (2.11)�(2.15) be satis�ed. It
follows from (2.11)�(2.14) that
H
�
��
q;n
(1) 0
0 ��
q;n
(1)
�
J
�
�q;n(1) 0
0 �q;n(1)
�
H
= H
�
Dq;n 0
0 Dq;n
�
+
�
D�
q;n 0
0 D�
q;n
�
H �H;
where Dq;n = Iq + Vq;n + V 2
q;n
+ : : :+ V n
q;n
: Hence (2.18) is true. Relations (2.15)
and (2.17) implies that H � 0. With regard to (2.18) we obtain that B(�) is a
J-elementary multiple factor. Using again (2.18), we conclude that the matrix H
is determined uniquely by B(�). Therefore C and P are also determined uniquely
by B(�). Theorem 2.2 is proved.
Let bj = �
�Iq 0
0 Iq
�
: De�ne
bB(�) = bjB��1
��
�bj = bj �d�0 + d�1� + : : :+ d�n+1�
n+1
�bj = bd0+ bd1�+ : : :+ bdn+1�n+1:
(2:29)
Since B(�) is a J-elementary multiple factor with the pole of multiplicity n + 1
at the point � = 0, then bB(�) is a J-elementary multiple factor with the pole of
multiplicity n + 1 at the point � = 1. In terms of the J -elementary multiple
factor bB(�) Theorem 2.2 can be reformulated in the following way.
Theorem 2.3. 1) Let bB(�) be a J-elementary multiple factor of the form
(2.29). There exist unique matrices P , C 2 C
q�q satisfying conditions (2.11)�
(2.15). Moreover, bB(�) can be represented in the following form:
bB(�) = I + (1� �)bj � �q;n(�) 0
0 �q;n(�)
�
H
�
��
q;n
(1) 0
0 ��q;n(1)
�bjJ; (2:30)
where H has the form (2.17).
2) Let conditions (2.11)�(2.15) be satis�ed and let bB(�) be the matrix-valued
function of the form (2.30). Then bB(�) is a J-elementary multiple factor. More-
over, bB(�)J bB�(�)� J
= (1� j � j2)bj � �q;n(�) 0
0 �q;n(�)
�
H
�
��
q;n
(�) 0
0 ��q;n(�)
�bj; (2:31)
where �q;n(�) has the form (2.5).
234 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
3. Solving of the basic matricial inequalities
Let E be a unitary space of dimension q and [E] be the set of linear oper-
ators acting on E. Denote by C[E] the class of operator-valued functions F(�)
analytical in D , such that for all � 2 D we have F(�) 2 [E] and ReF(�) =
1
2
(F(�) + F�(�)) � 0: Let an orthonormal basis in E be �xed. We identify
matrix-valued functions F(�) 2 Cq of the form (1.1) and corresponding to them
operator-valued functions from C[E].
Taking into account the block structure of the matrix Cn (see (1.2)), we con-
clude that the block matrix An of the form (1.3) can be considered as an operator
acting in the space
E(n) = E �E � : : :�E| {z }
n+1
:
We embed E(k�1) into E(k) in the following way E(k) = E(k�1)�E, k = 1; 2; : : : ; n,
E(0) = E.
Subspace of the type K introduced in [12] plays an important role when we
solve the degenerate Carath�eodory problem. In the case of the Carath�eodory
problem this subspace is de�ned in the following way.
De�nition 3.1. A subspace L � E(n) is said to be a subspace of the type K if:
1) L is the complement to the kernel of An, i.e., LuKerAn = E(n);
2) L is an invariant with respect to V �
q;n
, where Vq;n is de�ned by equality (2.9).
Note that in the case of the degenerate Carath�eodory problem the existence
of a subspace of the type K for the matrix An of the form (1.3) is proved in the
same way as in the case of the degenerate matricial Schur problem [12].
Let L be an arbitrary subspace of the type K, let P = PL be the orthoprojec-
tor onto L and C = PCn, where Cn has the form (1.2). With regard to de�ni-
tion 3.1, taking into account properties of the orthoprojector P and the equality
PC�+CP = PAnP , we obtain, that conditions (2.11)�(2.15) are satis�ed for the
matrices P and C.
The operator eAn = (PC� + CP ) jL = PAnP jL : L ! L is a nondegenerate
operator. The orthogonal decomposition E(n) = L�L? allows us to consider the
block representation
An =
� eAn B
B� D
�
=
�
I 0
X� I
� � eAn 0
0 0
� �
I X
0 I
�
; (3:1)
where X is a solution of the equation eAnX = B. Using this decomposition
E(n), we conclude that the operator PC� + CP has the form
� eAn 0
0 0
�
. Then
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 235
N.N. Chernovol
(PC� + CP )[�1] has the following block representation
� eA�1n 0
0 0
�
. It follows
from (3.1) that KerAn = �
�
�X
I
�
. Hence Ker [�X�; I] = �An
:
Now let us solve the matricial inequalities (1.4) and (1:40). Rewritten them in
the form "
An B(�; n)
� F(�)+F�(�)
1�� ��
#
� 0; � 2 D (3:2)
and "
An eB(�; n)
� F(�)+F�(�)
1�� ��
#
� 0; � 2 D nf0g (3:20)
respectively. Here
B(�; n) = ��q;n(�)F
�(�) + Cn�
�
q;n(�);
eB(�; n) = 1
�
��
q;n
�
1
��
�
F (�)�
1
�
Cn�
�
q;n
�
1
��
�
:
With regard to [12, p. 48, 49] we can show that inequality (3.2) holds if and only
if
[�X�; I]B(�; n) = 0; � 2 D ; (3:3)
F(�) + F�(�)
1� j � j2
�B�(�; n) (PC� + CP )[�1]B(�; n) � 0; � 2 D ; (3:4)
and that inequality (3:20) holds if and only if
[�X�; I] eB(�; n) = 0; � 2 D nf0g; (3:30)
F(�) + F�(�)
1� j � j2
� eB�(�; n) (PC� + CP )[�1] eB(�; n) � 0; � 2 D nf0g: (3:40)
Consider (3.3), (3.4). Inequality (3.4) may be solved as in the nondegenerate case
(see [2, 12]). Since
F(�) + F�(�)
1� j � j2
= [F(�); I]
J
1� j � j2
�
F�(�)
I
�
;
B(�; n) = [I; Cn]
�
��q;n(�) 0
0 ��q;n(�)
� �
F�(�)
I
�
;
then (3.4) can be represented in the form
[F(�); I]
�
J
1� j � j2
236 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
�J
�
�q;n(�) 0
0 �q;n(�)
�
H
�
��q;n(�) 0
0 ��
q;n
(�)
�
J
��
F�(�)
I
�
� 0; � 2 D ;
(3:5)
where
H =
�
C�
P
�
(PC� + CP )[�1] [C;P ] :
Since for the matrices P and C conditions (2.11)�(2.15) are ful�lled, then
Theorem 2.2 implies that
B(�) = I +
1� �
�
J
�
�q;n(1) 0
0 �q;n(1)
�
H
"
��
q;n
(1�� ) 0
0 ��
q;n
(1�� )
#
is a J-elementary multiple factor. In addition, equality (2.18) take place. We
have
B�1(�) = JB�(
1
��
)J; � 6= 0;
because B(�) is a J-unitary on the boundary of the unit disk. It follows from (2.18)
we get
J � B�1(�)JB��1(�)
1� j � j2
= J
�
�q;n(�) 0
0 �q;n(�)
�
H
�
��q;n(�) 0
0 ��
q;n
(�)
�
J:
By substitution the last expression into (3.5), we obtain
[F(�); I]B�1(�)JB��1(�)
�
F�(�)
I
�
� 0; � 2 D : (3:6)
Let us de�ne the pair of the matrix-functions [u(�); v(�)] in the following way:
[u(�); v(�)] = [F(�); I]B�1(�); � 2 D : (3:7)
In a way analogous to that used in the nondegenerate case (see [2, 12]) we prove
that the pair [u(�); v(�)] satis�es the following conditions:
(1) the matrix-functions u(�), v(�) are analytical in D ;
(2) the pair [u(�); v(�)] is a J-nonexpansive pair in D , i.e., for each � 2 D
[u(�); v(�)]J
�
u�(�)
v�(�)
�
� 0; (3:8)
(3) for all � 2 D the following inequality holds
[u(�); v(�)]
�
u�(�)
v�(�)
�
> 0; (3:9)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 237
N.N. Chernovol
i.e., the pair [u(�); v(�)] is nonsingular in D .
Let us decompose B(�) into the blocks according to the block representa-
tion (2.3) of matrix J
B(�) =
�
a(�) b(�)
c(�) d(�)
�
:
Then using (3.7), we obtain
F(�) = u(�)a(�) + v(�)c(�); I = u(�)b(�) + v(�)d(�):
Hence,
F(�) = [u(�)b(�) + v(�)d(�)]�1[u(�)a(�) + v(�)c(�)]: (3:10)
The converse is also true. Let [u(�); v(�)] be an arbitrary nonsingular J-nonexpan-
sive pair of analytic matrix-functions in D . Then the matrix u(�)b(�) + v(�)d(�)
is invertible in D and F(�) satis�es condition (3.6). Hence it also satis�es inequa-
lity (3.4). Thus, the following lemma is proved.
Lemma 3.1. The general solution F(�) of inequality (3.4) is represented
in the form of the linear fractional transformation (3.10), where the parameter
[u(�); v(�)] is a nonsingular J-nonexpansive pair of analytical matrix-functions in
D . The J-elementary multiple factor
B(�) =
�
a(�) b(�)
c(�) d(�)
�
of the form (2.16) with the pole of multiplicity n + 1 at the point � = 0 is the
matrix of coe�cients of the linear fractional transformation. In (2.16) P is the
orthoprojector onto one of subspaces of the type K and C = PCn.
Recall, that nonsingularity (J-nonexpansibility respectively) in D of the pair of
matrix-functions
� bu(�)bv(�)
�
means nonsingularity (J-nonexpansibility respectively)
in D of the pair [bu�(�); bv�(�)].
Analogously to Lemma 3.1 due to Theorem 2.3 we obtain
Lemma 3.2. The general solution F(�) of inequality (3:40) is represented in
the form of the linear fractional transformation:
F(�) = [ba(�)bu(�) +bb(�)bv(�)][bc(�)bu(�) + bd(�)bv(�)]�1; (3:100)
where the parameter
� bu(�)bv(�)
�
is a nonsingular J-nonexpansive pair of analytical
matrix-functions in D . The J-elementary multiple factor
bB(�) = " ba(�) bb(�)bc(�) bd(�)
#
238 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
of the form (2.30) with the pole of multiplicity n+ 1 at the point � =1 is a the
matrix of coe�cients of the linear fractional transformation. In (2.30) P is the
orthoprojector onto one of subspaces of the type K and C = PCn.
Now let us choose solutions satisfying condition (3.3) ((3:30) respectively) in
the set of solutions of the form (3.10) ((3:100) respectively) of inequality (3.4)
((3:40) respectively) .
Let P0 be an orthoprojector in E(n) onto KerAn: Repeating the reasonings of
the paper [12, p. 51, 52], we get that condition (3.3) is equivalent to the condition
P0[I; Cn]
�
��q;n(1) 0
0 ��
q;n
(1)
� �
u�(�)
v�(�)
�
= 0; � 2 D ; (3:11)
and (3:30) is equivalent to the condition
P0[I;�Cn]
�
��q;n(1) 0
0 ��q;n(1)
� � bu(�)bv(�)
�
= 0; � 2 D ; (3:110)
respectively. Equalities (3.11) and (3:110) we can rewrite in the form
u(�)�q;n(1)P0 � v(�)�q;n(1)CnP0 = 0; � 2 D ; (3:12)
bu�(�)�q;n(1)P0 + bv�(�)�q;n(1)CnP0 = 0; � 2 D ; (3:120)
respectively.
Note, that ��
q;n
(1)�q;n(1) = Fq;n + F �
q;n
+ I; where
Fq;n =
2666664
0 : : : : : : : : : 0
Iq 0 : : : : : : 0
Iq Iq 0 : : : 0
...
...
. . .
. . .
...
Iq Iq : : : Iq 0
3777775 :
Taking into account Fq;nCn = CnFq;n and An = Cn + C�n, we obtain
P0C
�
n�
�
q;n(1)�q;n(1)P0 + P0�
�
q;n(1)�q;n(1)CnP0
= P0
�
C�n(Fq;n + F �q;n + I) + (Fq;n + F �q;n + I)Cn
�
P0
= P0(AnFq;n + F �q;nAn +An)P0 = 0;
i.e.,
P0C
�
n�
�
q;n(1)�q;n(1)P0 + P0�
�
q;n(1)�q;n(1)CnP0 = 0: (3:13)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 239
N.N. Chernovol
Consider the operators r = �P0C
�
n
��
q;n
(1) and s = P0�
�
q;n
(1) acting in E on
KerAn. Then (3.13) can be rewritten in the form
[r; s]J
�
r�
s�
�
= 0; (3:14)
where J has the form (2.3).
Let eJ =
�
�Iq 0
0 Iq
�
and � = 1p
2
�
�Iq Iq
Iq Iq
�
. Note that � = �� = ��1 and
�J�� = eJ . It follows from here that
[r; s]J
�
r�
s�
�
= [r; s]�� eJ� � r�
s�
�
=
1
2
[�r + s; r + s] eJ � �r� + s�
r� + s�
�
=
1
2
(�(�r + s)(�r� + s�) + (r + s)(r� + s�)) = 0;
i.e., condition (3.14) is equivalent to the equality
(�r + s)(�r� + s�) = (r + s)(r� + s�): (3:15)
Let M0 and N0 be the ranges of the operators (�r
� + s�) and r� + s� respec-
tively. Relation (3.15) allows us to de�ne the unitary operator U : M0 ! N0,
such that
U(�r� + s�) = r� + s�: (3:16)
Now let us rewrite condition (3.12) for the pair [u(�); v(�)] in terms of the operator
U . Note that (3.12) is equivalent to the equality
[u(�); v(�)]J
�
r�
s�
�
= [u(�); v(�)]�� eJ� � r�
s�
�
=
1
2
[�u(�) + v(�); u(�) + v(�)] eJ � �r� + s�
r� + s�
�
=
1
2
(�(�u(�) + v(�))(�r� + s�) + (u(�) + v(�))(r� + s�)) = 0; � 2 D :
Taking into account (3.16), we can rewrite this condition in the form
(�(�u(�) + v(�)) + (u(�) + v(�))U) (�r� + s�) = 0; � 2 D : (3:17)
From (3.8), (3.9) we obtain
(u(�) + v(�))(u�(�) + v�(�))
240 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
= (u(�)v�(�) + v(�)u�(�)) + (u(�)u�(�) + v(�)v�(�)) > 0; � 2 D :
Hence for all � 2 D the matrix u(�) + v(�) is invertible. Then (3.17) is equivalent
to the condition
(v(�) + u(�))�1(v(�)� u(�))jM0
= U; � 2 D : (3:18)
Hence (3.12) is also equivalent to this condition. Analogously, we obtain that
(3:120) is equivalent to the equality
(bv�(�) + bu�(�))�1(bv�(�)� bu�(�))jN0
= U�; � 2 D : (3:180)
Thus, the following statements are proved.
Theorem 3.1. Let fckg
n
k=0 � C
q�q , the matrix Cn have the form (1.2) and
let for the matrix An = Cn + C�
n
condition (1.3) hold. Then the general solution
F(�) of basic matricial inequality (1.4) is represented in the form of the linear
fractional transformation:
F (�) = [u(�)b(�) + v(�)d(�)]�1[u(�)a(�) + v(�)c(�)];
where the parameter [u(�); v(�)] is a nonsingular J-nonexpansive pair of analytical
matrix-functions [u(�); v(�)] in D and satis�es the condition
(v(�) + u(�))�1(v(�)� u(�))jM0
= U; � 2 D :
Here U is determined by the problem data from the equality
U(�q;n(1)CnP0 +�q;n(1)P0) = ��q;n(1)CnP0 +�q;n(1)P0;
where P0 is the orthoprojector onto KerAn, �q;n(�) has the form (2.5). Moreover,
U is a unitary mapping of M0 to N0, where M0 is the range of the operator
�q;n(1)CnP0 + �q;n(1)P0 and N0 is the range of the operator (��q;n(1)CnP0 +
�q;n(1)P0). The J-elementary multiple factor
B(�) =
�
a(�) b(�)
c(�) d(�)
�
of the form (2.16) with the pole of multiplicity n + 1 at the point � = 0 is the
matrix of coe�cients of the linear fractional transformation. In (2.16) P is the
orthoprojector onto one of subspaces of the type K and C = PCn.
Theorem 3.2. Let fckg
n
k=0 � C
q�q , the matrix Cn have the form (1.2) and
let for the matrix An = Cn + C�n condition (1.3) hold. Then the general solution
F(�) of dual matricial inequality (1:40) is represented in the form of the linear
fractional transformation:
F(�) = [ba(�)bu(�) +bb(�)bv(�)][bc(�)bu(�) + bd(�)bv(�)]�1;
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 241
N.N. Chernovol
where the parameter
� bu(�)bv(�)
�
is a nonsingular J-nonexpansive pair of analytical
matrix-functions in D and satis�es the condition
(bv�(�) + bu�(�))�1(bv�(�)� bu�(�))jN0
= U�; � 2 D :
Here the operator U is determined as in Theorem 3.1. The J-elementary multiple
factor bB(�) = " ba(�) bb(�)bc(�) bd(�)
#
of the form (2.30) with the pole of multiplicity n + 1 at the point � = 1 is the
matrix of coe�cients of the linear fractional transformation. In (2.30) P is the
orthoprojector onto one of subspaces of the type K and C = PCn.
R e m a r k 3.1. Assume that there exists a point �0 2 D such that matrix
v(�0) is invertible. Then analyticity of v(�) in D implies invertibility of the matrix
v(�) everywhere in D excepting, may be, some set G of isolated in D points.
Let !(�) = v�1(�)u(�), � 2 D nG. From (3.8) it follows that
Re!(�) =
1
2
(v�1(�)u(�) + u�(�)(v�1(�))�)
=
1
2
v�1(�)(u(�)v�(�) + v(�)u�(�))(v�1(�))� � 0; � 2 D nG:
Therefore the matrix I + !(�) is invertible for all � 2 D nG and the function
s(�) = (I + !(�))�1(I � !(�)) (3:19)
is analytical in D nG and satis�es to the condition (see, e.g., [5, point 1.3])
ks(�)k � 1; � 2 D nG: (3:20)
Relation (3.19) is equivalent to
!(�)(I + s(�)) = I � s(�); � 2 D nG:
Hence the matrix I + s(�) is invertible for all � 2 D nG and the representation
!(�) = (I � s(�))(I + s(�))�1; � 2 D nG; (3:21)
is valid.
Denote by Sq the set of matrix-valued functions S(�) analytical in D with
values in C
q�q and satisfying the inequality kS(�)k � 1 for all � 2 D . From (3.20)
we conclude, that all points of set G are removable singular points for the matrix-
function s(�). Extending the s(�) to the points of G by continuity, we obtain
242 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
The degenerate Carath�eodory problem and the elementary multiple factor
function S(�) 2 Sq. From (3.21) it follows that
(�) = (I � S(�))(I + S(�))�1
belongs to the class Cq and it is the extension of the matrix-function !(�), � 2
D nG, to D .
Now (3.18) can be rewritten
(I +
(�))�1(I �
(�))jM0
= U; � 2 D : (3:22)
Thus, if there exists a point �0 2 D such that the matrix v(�0) is invertible, then
the corresponding solution F(�) of Carath�eodory problem (see Theorem 3.1) is
represented in the form of the linear fractional transformation F(�) = [
(�)b(�)+
d(�)]�1[
(�)a(�) + c(�)] of the matrix-function
(�) 2 Cq, satisfying condi-
tion (3.22). Moreover, this condition holds if and only if
(�) admits the repre-
sentation
(�) = (I � S(�))(I + S(�))�1;
where S(�) 2 Sq and S(�)jM0
= U , � 2 D . Note that from invertibility of the
matrix I + S(�), � 2 D it follows that �1 does not belong to the spectrum of the
operator U .
Analogous remark can also be made in the case of Theorem 3.2.
References
[1] I.V. Kovalishina, J-expansive matrix-valued functions in the Carath�eodory problem.
� Dokl. Akad. Nauk ArmSSR 59 (1974), No. 3, 129�135. (Russian)
[2] I.V. Kovalishina, Analytical theory of a class of interpolation problems. � Izv.
Akad. Nauk USSR 47 (1983), No. 3, 455�497. (Russian)
[3] C. Carath�eodory, �Uber den Variabilit�at der Koe�zienten von Potenzreihen, die
gegebene Werte nicht annehmen. � Mat. Ann. 64 (1907), 95�115.
[4] C. Carath�eodory, �Uber den Variabilit�at der Fourierschen Konstanten von positiven
harmonischen Funktionen. � Rendiconti del circolo matematico di Palermo XXXII
(1911), 193�217.
[5] V.K. Dubovoy, B. Fritzsche, and B. Kirstein, Matricial version of the classical Schur
problem. B.G. Teubner, Leipzig, 1992.
[6] B. Fritzsche and B. Kirstein, An extension problem for nonnegative hermitian block
Toeplitz matrices. IV. � Math. Nachr. 143 (1989), 329�354.
[7] I.V. Kovalishina and V.P. Potapov, Inde�nite metric in the Nevanlinna�Pick prob-
lem. � Dokl. Akad. Nauk ArmSSR 59 (1974), No. 1, 17�22. (Russian)
[8] I.V. Kovalishina, j-expansive matrix-valued functions and the classical moment
problem. � Dokl. Akad. Nauk ArmSSR 60 (1975), No. 1, 3�10. (Russian)
[9] V.P. Potapov, General theorems on the structure and the splitting of elementary
factors of analytical matrix-valued functions. � Dokl. Akad. Nauk ArmSSR 48
(1969), No. 5, 257�263. (Russian)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 243
N.N. Chernovol
[10] A.V. E�mov and V.P. Potapov, J-expansive matrix-valued functions and their
role in the analytical theory of electrical circuits. � Usp. Mat. Nauk 33 (1973),
No. 1(169), 65�130. (Russian)
[11] I.V. Kovalishina, Additive decomposition of a reactive matrix-valued functions. �
Izv. Akad. Nauk ArmSSR 6 (1971), No. 1, 43�60. (Russian)
[12] V.K. Dubovoy, Inde�nite metric in the interpolation Schur problem for analytic
functions. IV. � Teor. Funkts. Funkts. Anal. i ikh Prilozhen. 42 (1984), 46�57.
(Russian)
[13] L.A. Galstjan, Analytical J-expansive matrix-valued functions and the Schur prob-
lem. � Izv. Akad. Nauk ArmSSR 12 (1977), No. 3, 204�228. (Russian)
[14] V.K. Dubovoy, Parametrization of elementary multiple factors of nonfull rank. �
Analysis in in�nite-dimensional spaces and operator theory. Naukova Dumka, Kiev
(1983), 54�68. (Russian)
244 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
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| id | nasplib_isofts_kiev_ua-123456789-106575 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T18:36:32Z |
| publishDate | 2005 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Chernovol, N.N. 2016-09-30T18:04:52Z 2016-09-30T18:04:52Z 2005 The degenerate Carathéodory problem and the elementary multiple factor / N.N. Chernovol // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 2. — С. 225-244. — Бібліогр.: 14 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106575 The degenerate matricial interpolation Caratheodory problem is solved. To solve this problem we use the V.P. Potapov's approach based on the theory of J-expansive matrix-functions. The K-type subspace technique also plays an important role in these investigations. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии The degenerate Carathéodory problem and the elementary multiple factor Article published earlier |
| spellingShingle | The degenerate Carathéodory problem and the elementary multiple factor Chernovol, N.N. |
| title | The degenerate Carathéodory problem and the elementary multiple factor |
| title_full | The degenerate Carathéodory problem and the elementary multiple factor |
| title_fullStr | The degenerate Carathéodory problem and the elementary multiple factor |
| title_full_unstemmed | The degenerate Carathéodory problem and the elementary multiple factor |
| title_short | The degenerate Carathéodory problem and the elementary multiple factor |
| title_sort | degenerate carathéodory problem and the elementary multiple factor |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106575 |
| work_keys_str_mv | AT chernovolnn thedegeneratecaratheodoryproblemandtheelementarymultiplefactor AT chernovolnn degeneratecaratheodoryproblemandtheelementarymultiplefactor |