Triangular models of commutative systems of linear operators close to unitary operators
Triangular models are constructed for commutative systems of linear bounded operators close to unitary operators. The construction of these models is based on the continuation of basic relations for the characteristic function along the general chain of invariant subspaces.
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2016
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507752609087488 |
|---|---|
| author | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. |
| author_facet | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. |
| author_sort | Hatamleh, R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:30:15Z |
| description | Triangular models are constructed for commutative systems of linear bounded operators close to unitary operators. The construction of these models is based on the continuation of basic relations for the characteristic function along the general chain of invariant subspaces. |
| first_indexed | 2026-03-24T02:14:19Z |
| format | Article |
| fulltext |
UDC 517.9
R. Hatamleh (Jadara Univ., Irbid-Jordan),
V. A. Zolotarev (Inst. Low Temperature Phys. and Eng. Nat. Acad. Sci. Ukraine; V. N. Karazin Kharkiv Nat. Univ.)
TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS
OF LINEAR OPERATORS CLOSE TO UNITARY ONES
ТРИКУТНI МОДЕЛI КОМУТАТИВНИХ СИСТЕМ ЛIНIЙНИХ ОПЕРАТОРIВ,
БЛИЗЬКИХ ДО УНIТАРНИХ
Triangular models are constructed for commutative systems of linear bounded operators close to unitary operators. The
construction of these models is based on the continuation of basic relations for the characteristic function along the general
chain of invariant subspaces.
Побудовано трикутнi моделi комутативних систем лiнiйних обмежених операторiв, близьких до унiтарних. Побудо-
ву цих моделей засновано на подовженнi основних спiввiдношень для характеристичної функцiї вздовж загального
ланцюжка iнварiантних просторiв.
Introduction. It is common to consider [2, 3, 5] triangular or functional model as an analogue of
spectral decomposition for nonself-adjoint and nonunitary operators. For the first time, a triangular
model for nonself-adjoint operator has been built by M. S. Livšic, for nonunitary operator — by
V. T. Polyatsky [2]. Triangular models for commutative systems of linear bounded nonself-adjoint
operators has been built by V. A. Zolotarev [4], V. Vinnikov (part IV of [5]). These models are
based on the basic idea of M. S. Livšic [5] of the spectral analysis of this class of operator systems.
Reasonable constructions for the commutative systems of the nonunitary operators were built in [6, 7].
These constructions are the development of the method of M. S. Livšic [5]. This paper is dedicated
to the construction of triangular models for commutative systems \{ T1 , T2\} of linear operators close
to unitary ones. Note that some results stated in this paper were announced in [1]. An analogue of
the Hamilton — Cayley theorem is an important corollary of the constructed model representations,
namely, it is proved that the polynomial \BbbP (z1, z2) with antiholomorphic involution with respect to
the unit circle is such that \BbbP (T1, T2) = 0. This result for commutative systems of nonself-adjoint
operators has been obtained earlier by M. S. Livšic [5].
1. Let T be a linear bounded operator in a Hilbert space H. Let us recall that the set
\bigtriangleup =
\biggl(
\sigma ; H \oplus E; V =
\biggl[
T \Phi
\Psi K
\biggr]
;H \oplus \~E; \~\sigma
\biggr)
, (1)
is called a unitary colligation [2, 3, 6] if the operator V : H \oplus E \rightarrow H \oplus \~E has properties
V \ast
\Biggl[
I 0
0 \~\sigma
\Biggr]
V =
\Biggl[
I 0
0 \sigma
\Biggr]
, (21)
V
\Biggl[
I 0
0 \sigma - 1
\Biggr]
V \ast =
\Biggl[
I 0
0 \~\sigma - 1
\Biggr]
, (22)
where \sigma and \~\sigma are self-adjoint invertible operators acting in the Hilbert spaces E and \~E respectively.
As it is known [2, 3, 6], for every bounded operator there always exists such unitary colligation \Delta
c\bigcirc R. HATAMLEH, V. A. ZOLOTAREV, 2016
694 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 695
(1), of which T is the main operator. The characteristic function S\Delta (z) of a colligation \Delta ,
S\Delta (z) = K +\Psi (zI - T ) - 1\Phi , (3)
is the main analytic object in terms of which the spectral analysis of a operator T is realized [2, 3, 6].
Let \mathrm{d}\mathrm{i}\mathrm{m}E = \mathrm{d}\mathrm{i}\mathrm{m} \~E = r < \infty , then it is possible to suppose that E = \~E; suppose also that
\sigma = \sigma = J where J is an involution (J = J = J - 1). The well-known result of V. P. Potapov in this
case (see [2, 3]) gives us the multiplicative decomposition of the characteristic function. Namely, the
characteristic function S\Delta (z) (3) has a representation
S\Delta (z) = U
\curvearrowleft
l\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
ei\varphi t + z
ei\varphi t - z
JdFt
\biggr\} \curvearrowleft
N\prod
k=1
\Bigl(
Rk - ei\varphi k (zI - \alpha k)
- 1 Jdk
\Bigr)
. (4)
Moreover, U is a J-unitary matrix, \varphi t is a nondecreasing function on [0, l], 0 \leq \varphi t \leq 2\pi (0 \leq l <
< \infty ), Ft is a nondecreasing matrix-function for which \mathrm{t}\mathrm{r}Ft \equiv t, and the matrices \alpha k, Rk and dk
are such that
1) \alpha k are J-normal matrices, \alpha k\alpha
+
k = \alpha +
k \alpha k (\alpha +
k = j\alpha \ast
kj);
2) dk = J - \alpha \ast
kJ \alpha k = j
\bigl(
I - R2
k
\bigr)
\geq 0;
3) Rk is a J-module of a matrix \alpha k, R
2
k = \alpha +
k \alpha k;
4) \alpha k, \alpha
+
k and Rk act on dkE
r (Er is a Euclidean space with dimension r) as a multiplication
by \mu k, \=\mu k, and | \mu k| respectively,
(\alpha k - \mu kI) dk =
\bigl(
\alpha +
k - \=\mu kI
\bigr)
dk = (Rk - | \mu k| I) dk = 0,
where \mu k /\in \BbbT , and \varphi k = \mathrm{a}\mathrm{r}\mathrm{g}\mu k (N \leq \infty ).
An arrow in (4) signifies multiplicativity [2] of the integral and product
\curvearrowleft
N\prod
k=1
ak = aNaN - 1 . . . a1,
\curvearrowleft
l\int
0
\mathrm{e}\mathrm{x}\mathrm{p}\{ N(t)dF (t)\} = \mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow \infty
dn\rightarrow 0
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl\{
N(\xi n)F (\Delta n)
\bigr\}
. . . \mathrm{e}\mathrm{x}\mathrm{p}
\bigl\{
N(\xi 1)F (\Delta 1)
\bigr\}
,
where ak, N(t), F (t) are matrices-functions and 0 = x0 < x1 < . . . < xn = l, \Delta xk = xk - xk - 1,
xk - 1 \leq \xi k \leq xk, dn = \mathrm{m}\mathrm{a}\mathrm{x}
k
\Delta xk.
Hereinafter, we consider the case when the characteristic function S\Delta (z) is given by
S\Delta (z) =
\curvearrowleft
l\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
ei\varphi t + z
ei\varphi t - z
JdFt
\biggr\}
, (5)
which means that the spectrum of T lies on the unit circle \BbbT . We introduce the space
L2
r,l(Fx) =
\left\{ f(x) = (f1(x), . . . , fr(x)) \in Er :
l\int
0
f(x)dFxf
\ast (x) < \infty
\right\} , (6)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
696 R. HATAMLEH, V. A. ZOLOTAREV
assuming that the respective factorization by the kernel of the metric has been done. Specify the
operator \r T in L2
r,l(Fx),
\Bigl(
\r Tf
\Bigr)
(x) = f(x)ei\varphi x - 2
l\int
x
f(t)dFt\Phi
\ast
t\Phi
\ast - 1
x Jei\varphi x , (7)
where the matrix-function \Phi x is the solution of the equation
\Phi x +
x\int
0
\Phi tdFtJ = I. (8)
Define now the operators
\r \Psi : L2
r,l(Fx) \rightarrow Er; \r \Psi f(x) = -
\surd
2
l\int
0
f(t)dFt\Phi
\ast
t ,
\r \Phi : Er \rightarrow L2
r,l(Fx); \r \Phi g(x) = g
\surd
2\Psi xe
i\varphi x ,
\r K : Er \rightarrow Er; \r K = S(\infty ),
(9)
in this case the matrix-function \Psi x satisfies the equation
\Psi x +
l\int
x
\Psi tdFtJ = J. (10)
It is easy to see [2] that the collection
\Delta c =
\biggl(
J ; L2
r,l(Fx)\oplus Er; \r VT =
\biggl[
\r T \r \Phi
\r \Psi \r K
\biggr]
; L2
r,l(Fx)\oplus Er; J
\biggr)
, (11)
is a unitary colligation, the characteristic function of which coincides with S(z) (5), and \r T , \r \Phi , \r \Psi ,
\r K have the form of (7), (9).
Let us recall that the colligation \Delta (1) is said to be simple [2, 3] if H1 = H,
H1 = \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}
\Bigl\{
Tn\Phi E + T \ast m\Psi \ast \~E : n,m \in \BbbZ +
\Bigr\}
. (12)
It is easy to show that the subspace H0 = H\ominus H1 reduces T and that the restriction T on H0 induces
the unitary operator [2, 3].
Theorem 1 [2]. When the spectrum of the operator T lies on the circle T, the simple unitary
colligation \Delta (1) is unitarily equivalent to the simple part of the unitary colligation \Delta c (11).
2. Following [6], we define an analogue of the unitary operator \Delta (1) for the commutative
system of linear bounded operators \{ T1, T2\} , [T1, T2] = 0
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 697
Definition. The collection
\Delta =
\biggl(
\Gamma ; \{ Ns\} 21 ; \{ \sigma s\}
2
1 ; \{ \tau s\}
2
1 ;H \oplus E;
\bigl\{
Vs,
+
V s
\bigr\} 2
1
;H \oplus \~E; \{ \~\tau s\} 21 ; \{ \~\sigma s\}
2
1 ;
\bigl\{
\~Ns
\bigr\} 2
1
; \~\Gamma
\biggr)
, (13)
is said to be the commutative unitary colligation of the operator system \{ T1, T2\} in H ([T1, T2] = 0)
if in the Hilbert spaces E and \~E there exist such operators \sigma s, \tau s, Ns, \Gamma and \~\sigma s, \~\tau s, \~Ns, \~\Gamma respectively
(\sigma s, \tau s and \~\sigma s, \~\tau s are self-adjoint, s = 1, 2), such that the mappings
Vs =
\Biggl[
Ts \Phi Ns
\Psi K
\Biggr]
: H \oplus E \rightarrow H \oplus \~E,
+
Vs =
\Biggl[
T \ast
s \Psi \ast \~N\ast
s
\Phi \ast K\ast
\Biggr]
: H \oplus \~E \rightarrow H \oplus E,
(14)
satisfy the following relations:
V \ast
s
\Biggl[
I 0
0 \~\sigma s
\Biggr]
Vs =
\Biggl[
I 0
0 \sigma s
\Biggr]
, s = 1, 2, (151)
+
V \ast
s
\Biggl[
I 0
0 \~\tau s
\Biggr]
+
Vs =
\Biggl[
I 0
0 \~\tau s
\Biggr]
, s = 1, 2, (152)
T2\Phi N1 - T1\Phi N2 = \Phi \Gamma , \~N1\Psi T2 - \~N2\Psi T1 = \~\Gamma \Psi , (153)
\~N2\Psi \Phi N1 - \~N1\Psi \Phi N1 = K\Gamma - \~\Gamma K, (154)
KNs = \~NsK, s = 1, 2. (155)
It is easy to show [6] that for an arbitrary commutative system of bounded operators \{ T1, T2\}
there always exists such an isometric expansion
\bigl\{
Vs,
+
V s
\bigr\}
(14) (the colligation \Delta (13)) that conditions
(15) hold. Denote by S1(z) the characteristic function of the colligation \Delta (13) corresponding to the
operator T1,
S1(z) = K +\Psi (zI - T1)
- 1\Phi N1. (16)
It is shown in [6] that in the case of invertibility of the operators N1 and \~N1 the totality\Bigl\{
S1(z);\sigma s; \tau s;Ns; \Gamma ; \~\sigma s; \~\tau s; \~Ns; \~\Gamma
\Bigr\}
s=1,2
, (17)
is the total set of invariants of the commutative operator system \{ T1, T2\} . The condition of invertibility
of N1 and \~N1 is the additional restriction on the system of commutative operators T1 and T2. The
last fact means that in the case of the simplicity of the colligation \Delta (13) (H = H1 (12)) set (17)
defines the operator system \{ T1, T2\} up to the unitary equivalency [6].
We suppose that the operators \sigma 1 and \~\sigma 1 coincide with the involution J. Denote by N, \~N, \gamma , \~\gamma ,
\sigma , \~\sigma , \tau , \~\tau the operators corresponding to N - 1
1 N2, \~N - 1
1
\~N2, N
- 1
1 \Gamma , \~N - 1
1
\~\Gamma , \sigma 2, \~\sigma 2, N
- 1
1 \~\tau 2 (N
\ast
1 )
- 1 ,
\~N - 1
1 \~\tau 2
\bigl(
\~N\ast
1
\bigr) - 1
respectively. Then the characteristic function S(z) of the operator T1 (16) satisfies
the following relations describing the commutative property of T1 and T2 in terms of the external
parameters (17) [6, 7],
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
698 R. HATAMLEH, V. A. ZOLOTAREV
S(z)
\Bigl(
\~Nz + \~\gamma
\Bigr)
= (Nz + \gamma )S(z), (181)
S(z)JS\ast (w) - J - (Nz + \gamma ) \{ S(z)JS\ast (w) - J\} (N\ast \=w + \gamma \ast ) =
= (I - z \=w) \{ S(z)\~\sigma S\ast (w) - \sigma \} , (182)
S\ast (\=z) JS ( \=w) - J -
\Bigl(
\~N\ast z + \~\gamma \ast
\Bigr)
\{ S\ast (\=z) JS ( \=w) - J\}
\Bigl(
\~N \=w + \gamma
\Bigr)
=
= (I - z \=w) \{ S\ast (\=z)\tau S ( \=w) - \~\tau \} . (183)
Simultaneous reduction of the commutative operator system \{ T1, T2\} to the triangular type means
continuation and conservation of the main relations (18) along some chain of joint invariant subspaces
for T1 and T2.
There is no corresponding parallel to the material of book [5] in the work. It is proposed to study
the properties of the characteristic function S1(z) (16) in the presence of the colligation relations (15).
The main point is that the conditions (18) express the fact that the operator T2 commutes with
the operator T1, this imposes the additional conditions on the characteristic function S1(z) (16) of
the operator T1.
3. Suppose that the characteristic function of the operator T1 is given by (5),
S(z) = Sl(z), Sx(z) =
\curvearrowleft
x\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
ei\varphi t + z
ei\varphi t - z
JdFt
\biggr\}
, (19)
where x \in [0; l] and \varphi t, Ft have corresponding properties (see Section 1). First, let us study
continuation of the condition (181) assuming that dFx = axdx, where ax is a nonnegative matrix-
function on [0, l] such that \mathrm{t}\mathrm{r} ax \equiv 1.
Theorem 2. Sx(z) (19), where dFx = axdx, satisfies the intertwining condition
Sx(z)
\Bigl(
\~Nz + \~\gamma
\Bigr)
= (Nxz + \gamma x)Sx(z), (20)
then and only then when Nx and \gamma x are the solutions of the equations
N \prime
x = - [Jax, Nx] , N0 = \~N, (211)
\gamma \prime x = [Jax, \gamma x] , \gamma 0 = \~\gamma , (212)
moreover, \bigl[
Jax,
\bigl(
Nxe
i\varphi x + \gamma x
\bigr) \bigr]
= 0, (22)
which means that \gamma \prime x = ei\varphi xN \prime
x \forall x \in [0, l].
Proof. Differentiate equality (20) assuming that dFx = axdx, then
ei\varphi x + z
ei\varphi x - z
JaxSx(z)( \~Nz + \~\gamma ) =
\biggl\{
zN \prime
x + \gamma \prime x + (zNx + \gamma x)
ei\varphi x + z
ei\varphi x - z
Jax
\biggr\}
Sx(z).
Again taking into account (20) and the invertibility of Sx(\infty ), we obtain
ei\varphi x + z
ei\varphi x - z
[Jax, (zNx + \gamma x)] = zN \prime
x + \gamma \prime x, (23)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 699
which, after equating the corresponding coefficients by zk, k = 0, 1, 2), gives us (211), (212), and
(22). And the initial conditions of the Cauchy problems (21) follow from (20) when x = 0.
To prove the sufficiency, we use (23), which is equivalent to (211), (212), and (22), and the
equation for Sx(z), then we have
d
dx
\{ (zNx + \gamma x)Sx(z)\} =
ei\varphi x + z
ei\varphi x - z
Jax (zNx + \gamma x)Sx(z). (24)
Taking into account that the function Sx(z)
\Bigl(
z \~N + \~\gamma
\Bigr)
satisfies similar equation, we get
d
dx
\Bigl\{
Sx(z)
\Bigl(
z \~N + \~\gamma
\Bigr)
- (zNx + \gamma x)Sx(z)
\Bigr\}
=
=
ei\varphi x + z
ei\varphi x - z
Jax
\Bigl\{
Sx(z)
\Bigl(
z \~N + \~\gamma
\Bigr)
- (zNx + \gamma x)Sx(z)
\Bigr\}
,
which proves (20) in view of the uniqueness of the solution of the Cauchy problem since\Bigl\{
Sx(z)
\Bigl(
z \~N + \~\gamma
\Bigr)
- (zNx + \gamma x)Sx(z)
\Bigr\}
= 0, when x = 0.
Theorem 2 is proved.
Now let us consider similar continuation of relation (182) along the given chain of the invariant
subspaces.
Theorem 3. The matrix-function Sx(z) (19) where dFx = axdx satisfies the relation
Sx(z)JS
\ast
x(w) - J - (zNx + \gamma x) \{ Sx(z)JS
\ast
x(w) - J\} (N\ast
x \=w + \gamma \ast x) =
= (I - z \=w) \{ Sx(z)\~\sigma S
\ast
x(w) - \sigma x\} , (25)
under conditions (211), (212) and (22), if and only if the relations
\sigma \prime
x = 2NxJaxJN
\ast
x - Jax\sigma x - \sigma xaxJ, \sigma 0 = \~\sigma , (261)\bigl\{
NxJ
\bigl(
N\ast
x + ei\varphi x\gamma \ast x
\bigr)
- \sigma x
\bigr\}
ax = 0, (262)
JaxJ +NxJaxJN
\ast
x - \gamma xJaxJ\gamma
\ast
x = \sigma xaxJ + Jax\sigma x, (263)
hold for all x \in [0; l].
Proof. To prove the necessity, we differentiate relation (25) and use the equalities (20) – (24),
then we get \biggl(
ei\varphi x + z
ei\varphi x - z
+
e - i\varphi x + \=w
e - i\varphi x - \=w
\biggr)
\{ JaxJ - (zNx + \gamma x) JaxJ ( \=wN\ast
x + \gamma \ast x)\} =
= (I - z \=w)
\biggl\{
ei\varphi x + z
ei\varphi x - z
Jax\sigma x + \sigma xaxJ
e - i\varphi x + \=w
e - i\varphi x - \=w
- \sigma \prime
x
\biggr\}
.
Now equating the coefficients by equal powers zk \=w s, k, s = 0, 1, we obtain relations (261) – (263).
To prove the sufficiency, we consider the following function:
\Psi x(z, w) = Sx(z)JS
\ast
x(w) - J -
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
700 R. HATAMLEH, V. A. ZOLOTAREV
- (zNx + \gamma x) \{ Sx(z)JS
\ast
x(w) - J\} (N\ast
x \=w + \gamma \ast x)+
+ (z \=w - I) \{ Sx(z)\~\sigma S
\ast
x(w) - \sigma x\} .
From (23), (26) it follows that \Psi x(z, w) satisfies the equation
d
dx
\Psi x(z, w) =
ei\varphi x + z
ei\varphi x - z
Jax\Psi x(z, w) + \Psi x(z, w)axJ
e - i\varphi x + \=w
e - i\varphi x - \=w
,
and, taking into account that \Psi 0(z, w) = 0, we obtain \Psi x(z, w) = 0 \forall x \in [0, l], this fact proves (25).
Theorem 3 is proved.
Continuation of relation (183) along the given chain of the invariant subspaces leads us to the
following statement.
Theorem 4. Suppose that the matrix-function Sx(z) (19), where dFx = axdx, satisfies the
intertwining condition (20), then
S\ast
x(\=z)JSx( \=w) - J -
\Bigl(
\~N\ast z + \~\gamma \ast
\Bigr)
\{ S\ast
x(\=z)JSx( \=w) - J\}
\Bigl(
\~N \~w + \~\gamma
\Bigr)
=
= (I - z \=w) \{ S\ast
x(\=z)\tau xSx( \=w) - \~\tau \} , (27)
is true then and only then, when
\tau \prime x = \tau xJax + axJ\tau x - 2N\ast
xaxNx, \tau 0 = \~\tau , (281)\bigl\{
N\ast
xJ
\bigl(
Nx + e - \varphi x\gamma x
\bigr)
- \tau x
\bigr\}
Jax = 0, (282)
ax +N\ast
xaxNx - \gamma \ast xax\gamma x = \tau xJax + axJ\tau x, (283)
for all x \in [0, l] .
Proof. Differentiating (27) and using (20), we have\biggl(
e - i\varphi x + z
e - i\varphi x - z
+
ei\varphi x + \=w
ei\varphi x - \=w
\biggr)
\{ ax - (zN\ast
x + \gamma \ast x) ax ( \=wNx + \gamma x)\} =
= (I - z \=w)
\biggl\{
\tau xJax
ei\varphi x + \=w
ei\varphi x - \=w
+ axJ\tau x
e - i\varphi x + z
e - i\varphi x - z
+ \tau \prime x
\biggr\}
.
Now in order to obtain (281) – (283) it is necessary to equate the corresponding coefficients by powers
zk \=ws, k, s = 0, 1. Now, if we consider the function
\Psi x(z, w) = S\ast
x(\=z)JSx( \=w) - J -
\Bigl(
\~N\ast
xz + \~\gamma \ast
\Bigr)
\{ S\ast
x(\=z)JSx( \=w) - J\} \times
\times
\Bigl(
\~N \=w + \~\gamma
\Bigr)
(z \=w - I) \{ S\ast
x (\=z) \tau xSx( \=w) - \~\tau \} ,
then in view of the last equation and taking into account the intertwining relation (20) it is easy to
verify that
d
dx
\Psi x(z, w) = 0, and using the initial data \Psi 0(z, w) = 0, we obtain \Psi x(z, w) = 0, this
proves the sufficiency of the statement.
Theorem 4 is proved.
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TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 701
4. Substituting equality (262) in relation (263) and using (223), we see that\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
JaxJ
\bigl(
Nxe
i\varphi x + \gamma x
\bigr) \ast
= JaxJ.
Thus the operator J
\bigl(
N\ast
xe
- i\varphi x + \gamma \ast x
\bigr)
J is isometric in the metric given by the operator ax. Therefore,
this equality and relation (262) on the subspace Er
x = axE
r we can write in the following form:\bigl\{ \bigl(
Nxe
i\varphi x + \gamma x
\bigr)
J
\bigl(
N\ast
xe
- i\varphi x + \gamma x
\bigr)
- J
\bigr\}
fx = 0,\bigl\{
NxJ
\bigl(
N\ast
x + e - i\varphi x\gamma \ast x
\bigr)
- \sigma x
\bigr\}
fx = 0,
(29)
where fx \in Er
x. Similarly, it follows from relations (28) that\bigl(
Nxe
i\varphi x + \gamma x
\bigr) \ast
ax
\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
= ax,
this means that Nxe
i\varphi x + \gamma x is isometric in the metric ax, thus\Bigl\{ \bigl(
Nxe
i\varphi x + \gamma x
\bigr) \ast
J
\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
J - I
\Bigr\}
fx = 0,\bigl\{
N\ast
xJ
\bigl(
Nx + e - i\varphi x\gamma x
\bigr)
J - \tau xJ
\bigr\}
fx = 0,
(30)
where fx \in Er
x in view of (282).
Let us turn to the solvability of the equation system (21), (22) which we can write in the form
of (23),
d
dx
(zNx + \gamma x) =
\biggl[
ei\varphi x + z
ei\varphi x - z
Jax, (zNx + \gamma x)
\biggr]
. (31)
Following P. Lax [8], in order to integrate the equation L\prime
x = [Ax, Lx] it is necessary to find an
“isometric” operator-function Vx, such that V \prime
x = AxVx, where in this case Vx realizes the equivalency
between Lx and L0. In our case, Vx = Sx(z) and the given equation L\prime
x = [Ax, Lx] leads us to the
intertwining condition (20),
zNx + \gamma x = Sx(z)
\Bigl(
\~Nz + \~\gamma
\Bigr)
S - 1
x (z),
since, by our supposition, the matrix Sx(z) is invertible when | z| \gg 1 (for example). Hence,
eigenvalues of the matrices Nx and \gamma x do not depend on x, and the root subspaces Lz of the bundle
\~Nz1 + \~\gamma corresponding to the number z2 under the action of the matrix-function Sx(z) passes into
the root subspace Lz(x) = Sx(z)Lz of the linear pencil Nxz1 + \gamma x.
Theorem 5. Suppose that Sx(z) (19) is invertible at some point z0 \in \BbbC , then the solutions Nx
and \gamma x of the Cauchy problems (211), (212) exist, moreover, equality (22) holds.
Invertibility of Sx(z0) for all x \in [0, l] follows from the J-theory of V. P. Potapov [2] with the
condition that S(z0) is an invertible matrix. Furthermore, since relations (261) – (263) and (281) –
(283) are equivalent to the equalities (25) and (27) correspondingly with the condition that Nx and
\gamma x satisfy (211) and (212), then the existence of the matrix-functions \sigma x and \tau x is obvious.
Theorem 6. Suppose that Nx and \gamma x are the solutions of the Cauchy problems (211), (212) for
which (22) holds and the matrix-function Sx(z) is invertible at one point z0 \in \BbbC at least, then the
solutions \sigma x of the relations (26) and \tau x of (28) respectively exist and are unique.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
702 R. HATAMLEH, V. A. ZOLOTAREV
5. Now we turn to the construction of the triangular models of the systems of operators T1, T2.
To do this, we consider in the Hilbert space L2
r,l(Fx) (6) the system of linear operators
\Bigl(
\r T1f
\Bigr)
(x) = f(x)ei\varphi x - 2
l\int
x
f(t)dFt\Phi
\ast
t\Phi
\ast - 1
x Jei\varphi x ,
\Bigl(
\r T2f
\Bigr)
(x) = f(x)J
\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
J - 2
l\int
x
f(t)dFt\Phi
\ast
t\Phi
\ast - 1
x Jei\varphi x ,
(32)
where \Phi x is the solution of the integral equation (7) and Nx, \gamma x are the solutions of the Cauchy
problems (21) respectively. Note that the operator \r T1 in (32) coincides with \r T (6). First of all, we
make sure that the operators (32) are commutative. To do this, we consider
F (x)ei\varphi x =
\Bigl[
\r T1, \r T2
\Bigr]
f(x) = - 2
l\int
x
f(t)J
\bigl(
Nte
i\varphi t + \gamma t
\bigr)
Jatdt\Phi
\ast
t\Phi
\ast - 1
x Jei\varphi x+
+4
l\int
x
dt
l\int
t
f(s)asds\Phi
\ast
s\Phi
\ast - 1
t NtJe
i\varphi tat\Phi
\ast
t\Phi
\ast - 1
x Jei\varphi x+
+2
l\int
x
f(t)atdt\Phi
\ast
t\Phi
\ast - 1
x \gamma xJe
i\varphi x + 2
l\int
x
f(t)ei\varphi tatdt\Phi
\ast
t\Phi
\ast - 1
x NxJe
i\varphi x -
- 4
l\int
x
dt
l\int
t
f(s)asds\Phi
\ast
s\Phi
\ast - 1
t Jei\varphi tat\Phi
\ast
t\Phi
\ast - 1
x NxJe
i\varphi x .
Using the equations (211), (212) and the fact that
\bigl(
\Phi \ast - 1
x
\bigr) \prime
= \Phi \ast - 1
x Jax, as a result of the elementary
calculations using the equality (22), it is easy to see that the function Fx satisfies the differential
equation F \prime
x = FxaxJ, and, since Fl = 0, then Fx \equiv 0, which proves the commutative properties of
the operators \r T1, \r T2 (32).
Using equality (22), it is easy to show that\Bigl(
\r T \ast
1 f
\Bigr)
(x) = f(x)ei\varphi x - 2
x\int
0
f(t)e - i\varphi tdFtJ\Phi
- 1
t \Phi \ast
x,\Bigl(
\r T \ast
2 f
\Bigr)
(x) = f(x)
\bigl(
Nxe
i\varphi x + \gamma x
\bigr) \ast - 2
x\int
0
f(t)e - i\varphi tdFtJN
\ast
t \Phi
- 1
t \Phi x.
(33)
To construct the expansions Vs and
+
Vs (14), we have to calculate the defect operators I - \r T \ast
s
\r Ts and
I - \r Ts
\r T \ast
s , s = 1, 2. First of all, note the well-known fact [2] that
\Bigl(
I - \r T \ast
1
\r T1
\Bigr)
f(x) = 2
l\int
0
f(t)dFt\Phi
\ast
tJ\Phi x.
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TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 703
We consider \Bigl(
I - \r T \ast
2
\r T2
\Bigr)
f(x) = f(x)
\Bigl\{
I - J
\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
J
\bigl(
Nxe
i\varphi x + \gamma x
\bigr) \ast \Bigr\}
+
+ 2
x\int
0
f(t)J
\bigl(
Nte
i\varphi t + \gamma t
\bigr)
Je - i\varphi tatJdtN
\ast \Phi - 1
t \Phi x+
+ 2
l\int
x
f(t)atdt\Phi
\ast
t\Phi
\ast - 1
x NxJe
i\varphi x
\bigl(
N\ast
xe
- i\varphi x + \gamma \ast x
\bigr)
-
- 4
x\int
0
dt
l\int
t
f(s)asds\Phi
\ast
s\Phi
\ast - 1
t NtJatJ\Phi
- 1
t \Phi x.
From equation (261) and the fact that
\bigl(
\Phi - 1
x
\bigr) \prime
= axJ\Phi
- 1
x , we conclude
d
dx
\bigl(
\Phi \ast - 1
x \sigma x\Phi
- 1
x
\bigr)
= 2\Phi \ast - 1
x NxJaxJN
\ast
x\Phi
- 1
x .
Using relation (262), the equalities (29), and integration by parts, we obtain
\Bigl(
I - \r T \ast
2
\r T2
\Bigr)
f(x) = 2
l\int
0
f(t)dFt\Phi
\ast
t \~\sigma \Phi x.
Hence, if we define the operator \r \Psi : L2
r,l(Fx) \rightarrow Er,
\r \Psi f(x) = -
\surd
2
l\int
0
f(t)dFt\Phi
\ast
t , \r \Psi \ast \xi = -
\surd
2\xi \Phi x, (34)
where \xi \in Er and f(x) \in L2
r,l(Fx), then we get \r T \ast
1
\r T1 + \r \Psi \ast J\r \Psi = I and \r T \ast
2
\r T2 + \r \Psi \ast \~\sigma \r \Psi = I.
Let us calculate other defect operators I - \r Ts
\r T \ast
s , s = 1, 2, then [2],
\Bigl(
I - \r T1
\r T \ast
1
\Bigr)
f(x) = 2
l\int
0
f(t)e - i\varphi tdFt\Psi
\ast
tJ\Psi xe
i\varphi x ,
where \Psi x is the solution of the integral equation (9) and \Psi x = \Phi \ast
l\Phi
\ast - 1
x J. Next we consider\Bigl(
I - \r T2
\r T \ast
2
\Bigr)
f(x) = f(x)
\Bigl\{
I -
\bigl(
Nxe
i\varphi x + \gamma x
\bigr) \ast
J
\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
J
\Bigr\}
+
+2
x\int
0
f(t)
\bigl(
Nte
i\varphi t + \gamma t
\bigr) \ast
at\Phi
\ast
tdt\Phi
- 1
x NxJe
i\varphi x+
+2
x\int
0
f(t)e - i\varphi tatdtJN
\ast
t \Phi
- 1
t \Phi xJ
\bigl(
N\ast
xe
i\varphi x + \gamma x
\bigr)
J -
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
704 R. HATAMLEH, V. A. ZOLOTAREV
- 4
l\int
x
dt
t\int
0
f(s)e - i\varphi sasdsJN
\ast
s\Phi
- 1
s \Phi tat\Phi
\ast
t\Phi
\ast - 1
x \Phi xJe
i\varphi x ,
since (\Phi xJ\Phi
\ast
x)
\prime = - 2\Phi xax\Phi x, then after the integration by parts in the last integral and taking into
account (22) and (30), we obtain
\Bigl(
I - \r T2
\r T \ast
2
\Bigr)
f(x) = 2
l\int
0
f(t)e - i\varphi tatdtJN
\ast
t J\Psi
\ast
tJ\Psi xJNxJe
i\varphi x+
+2
l\int
x
f(t)atJ\gamma
\ast
t J\Phi
\ast
tdt\Phi
\ast - 1
x NxJe
i\varphi x + 2
x\int
0
f(t)e - i\varphi tatJN
\ast
t \Phi
- 1
t dt\Phi xJ\gamma xJ.
Representing the first integral as the sum of the integrals on the segments [0, x] and [x, l], we have
\Bigl(
I - \r T2
\r T \ast
2
\Bigr)
f(x) = 2
x\int
0
f(t)e - i\varphi tatJN
\ast
t \Phi
- 1
t dt
\bigl\{
\Phi lJ\Phi
\ast
l\Phi
\ast - 1
x NxJe
i\varphi x +\Phi xJ\gamma xJ
\bigr\}
+
+2
l\int
x
f(t)e - i\varphi sat
\bigl\{
J\gamma \ast t\Phi
\ast
l + JN\ast
t \Phi
- 1
t \Phi lJ\Phi
\ast
l e
- i\varphi t
\bigr\}
dt\Phi \ast - 1
x NxJe
i\varphi x . (35)
Since the above integrals are adjoint one to each other in terms of the metric L2
r,l(Fx), it is sufficient
to calculate one of them, for example, the first. It follows from (211) that
\bigl(
N\ast
t \Phi
- 1
t
\bigr) \prime
= atJN
\ast
t \Phi
- 1
t ,
therefore taking into account the initial data we get
N\ast
t \Phi
- 1
t = \Phi - 1
t
\~N\ast = J\Psi \ast
t\Phi
- 1
l
\~N\ast ,
and hence the first of the integrals in (35) is equal to
2
l\int
x
f(t)e - i\varphi sat\Psi
\ast
tdt\Phi
- 1
l
\~N\ast \bigl\{ \Phi lJ\Phi
\ast
l\Phi
\ast - 1
x NxJe
i\varphi x +\Phi xJ\gamma xJ
\bigr\}
.
Lemma. Suppose that \tau s satisfies the relations (28) and the matrices Nx and \gamma x are such that
the equalities (21) are true and, moreover, (29) takes place, then
\Phi - 1
l
\~N\ast \bigl\{ \Phi lJ\Phi
\ast
l\Phi
\ast - 1
x NxJ + e - i\varphi x\Phi xJ\gamma xJ
\bigr\}
= \tau l\Psi x, (36)
where \tau l is the value of the solution \tau x of problem (281) at the point x = l.
Sum of the integrals (35) with (36) can be represented in the form of
\Bigl(
I - \r T2
\r T \ast
2
\Bigr)
f(x) = 2
l\int
0
f(t)e - i\varphi tdFt\Psi
\ast
t \tau l\Psi xe
i\varphi x .
So, if we define the operator \r \Phi from Er into L2
r,l(Fx) by the formula coinciding with (8),
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TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 705
\r \Phi \xi =
\surd
2\xi \Psi xe
i\varphi x ,
\Bigl(
\r \Phi \ast f
\Bigr)
(x) =
\surd
2
l\int
0
f(t)e - i\varphi tat\Psi
\ast
t dt, (37)
where \xi \in Er, then we get \r T \ast
1
\r T \ast
1 +\r \Phi J\r \Phi \ast = I and \r T2
\r T \ast
2 +\r \Phi \tau l\r \Phi
\ast = I.
Proof. It follows from the Cauchy problem for Nx, (211) that Nx = \Phi \ast
x
\~N\Phi - 1
x or \Phi \ast - 1
x Nx =
= \~N\Phi \ast - 1
x and so
\Phi - 1
l
\Bigl\{
\~N\ast \Phi lJ\Phi
\ast
l
\~N\Phi \ast - 1
x J + e - i\varphi x \~N\ast \Phi xJ\gamma xJ
\Bigr\}
= \tau l\Psi x.
Taking the limit as z, \=w \rightarrow \infty in equality (27) and assuming that x = l, we obtain that \~N\ast (\Phi \ast
l J\Phi l -
- J) \~N + \~\tau = \tau l. Therefore
\Phi - 1
l
\Bigl\{ \Bigl(
\~N\ast J \~N - \~\tau
\Bigr)
\Phi \ast - 1
x J + e - i\varphi x \~N\ast \Phi xJ\gamma xJ
\Bigr\}
= 0,
and taking into account again that \~N\ast \Phi x = \Phi xN
\ast
x we find that
\~N\ast J \~N\Phi \ast - 1
x - \~\tau \Phi \ast - 1
x + e - i\varphi x\Phi xN
\ast J\gamma x = 0.
And using (30) N\ast
xJ\gamma xe
- i\varphi x = \tau x - N\ast
xJNx we have
\~N\ast J \~N\Phi \ast - 1
x - \~\tau \Phi \ast - 1
x +\Phi x (\tau x - N\ast
xJNx) = 0.
Thus
\~N\ast J \~N - \~\tau +\Phi x\tau x\Phi
\ast
x - \~N\ast \Phi xJ\Phi
\ast
x
\~N = 0.
Now to complete the proof of the lemma, it is to be noted that the last equality follows from (27)
after taking the limit as z, \=w \rightarrow \infty , with fixed x \in [0, l].
Lemma is proved.
Now we can construct the mappings
\circ
V1 =
\Biggl[
\r T1
\r \Phi
\r \Psi \r K
\Biggr]
,
\circ
V2 =
\Biggl[
\r T2
\r \Phi Nl
\r \Psi \r K
\Biggr]
,
\circ +
V1 =
\circ
V \ast
1 ,
\circ +
V2 =
\Biggl[
\r T \ast
2
\r \Psi \ast \~N\ast
\r \Phi \ast \r K\ast
\Biggr]
, (38)
in L2
r,l(Fx)\oplus Er where N
l
is the value of the solution Nx (211) at the point x = l; \r K = Sl(\infty ) = \Phi \ast
l
and the operators \r \Phi , \r \Psi are defined by the formulas (37) and (34) respectively. It is not difficult to
see [2] that
\circ
V \ast
1
\Biggl[
I 0
0 J
\Biggr]
\circ
V1 =
\Biggl[
I 0
0 J
\Biggr]
,
\circ
V1
\Biggl[
I 0
0 J
\Biggr]
\circ
V \ast
1 =
\Biggl[
I 0
0 J
\Biggr]
.
In order to verify that the analogous relations are valid for
\circ
V2 and
\circ +
V2, it is necessary to show that
\r T \ast
2
\r \Phi Nl + \r \Psi \ast \~\sigma \r K = 0, \r T \ast
2
\r \Psi \ast \~N\ast +\r \Phi \tau l\r K
\ast = 0. (39)
To prove the first relation in (39), we consider
\r T \ast
2 \xi Nl\Psi xe
i\varphi x = \xi Nl\Psi x
\bigl(
N\ast
x + ei\varphi x\gamma \ast x
\bigr)
- 2
x\int
0
\xi Nl\Psi tatJN
\ast
t \Phi
- 1
t dt\Phi x,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
706 R. HATAMLEH, V. A. ZOLOTAREV
where \xi \in Er. Then taking into account that
\bigl(
\Psi tN
\ast
t \Phi
- 1
l
\bigr) \prime
= 2\Psi tatJN
\ast
t \Phi
- 1
l we obtain that
\r T \ast
2 \xi Nl\Psi xe
i\varphi x = \xi Nl\Psi x\gamma
\ast
xe
i\varphi x + \xi Nl\Psi 0
\~N\ast \Phi x.
Since \Psi x = \Phi \ast
l\Phi
\ast - 1
x J and Nx\Phi
\ast
x = \Phi \ast
x
\~N, we have
\r T \ast
2 \xi Nl\Psi xe
i\varphi x = \xi \Psi xJNlJ\gamma
\ast
xe
i\varphi x + \xi Nl\Psi 0
\~N\ast \Phi x,
and in view of (29) we obtain the equality
\r T \ast
2 \xi Nl\Psi xe
i\varphi x = \xi \Psi xJ (\sigma x - NxJN
\ast
x) + \xi Nl\Psi 0
\~N\ast \Phi x.
Using the relation Nx(\Phi
\ast
xJ\Phi x - J)N\ast
x = \Phi \ast
x\~\sigma \Phi x - \sigma x, that follows from (25) as a result by taking
the limit as z, \=w \rightarrow \infty and the fact that Nx\Phi
\ast
x = \Phi \ast
x
\~N, \Psi 0 = \Psi \ast
l J, we finally get
\r T \ast
2 \xi Nl\Psi xe
i\varphi x = \xi \Phi \ast
x\~\sigma \Phi x,
this proves the necessity in view of the definition of the operators \r \Psi (34), \r \Phi (37) and \r K = \Phi \ast
l .
Let us prove the second equality in (39),
\r T \ast
2 \xi \~N\ast \Phi x = \xi \~N\ast \Phi xJ
\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
J - 2
l\int
x
\xi \~N\ast \Phi tat\Phi
\ast
tdt\Phi
\ast - 1
x NxJe
i\varphi x ,
where \xi \in Er, and since (\Phi tJ\Phi
\ast
t )
\prime = 2\Phi tat\Phi
\ast
t , then
\r T \ast
2 \xi
\~N\ast \Phi x = \xi \~N\ast \Phi xJ\gamma xJ + \xi \~N\ast \Phi lJ\Phi
\ast
l\Phi
\ast - 1
x NxJe
i\varphi x .
To prove the necessary equality \r T \ast
2 \xi
\~N\ast \Phi x = \xi \Phi l\tau l\Psi xe
i\varphi x , we have to prove that
\~N\ast \Phi xJ\gamma xJe
- i\varphi x + \~N\ast \Phi xJ\Psi xJNxJ = \Phi l\tau l\Psi x.
Now using the fact that \~N\ast \Phi x = \Phi xN
\ast
x and in view of (30), we obtain
\Phi x (\tau x - N\ast
xJNx) J + \~N\ast \Phi xJ\Psi xJNxJ = \Phi l\tau l\Psi x.
As a result of the multiplication from the right by J\Phi \ast
x, we get
\Phi x (\tau x - N\ast
xJNx) \Phi
\ast
x + \~N\ast \Phi lJ\Phi
\ast
l
\~N = \Phi l\tau l\Phi
\ast
l .
Taking the limit as z, \=w \rightarrow \infty in equality (27), we have
\~\tau - \~N\ast J \~N = \Phi x\tau x\Phi
\ast
x - \~N\ast \Phi xJ\Phi
\ast
x
\~N.
This means that
\~\tau - \~N\ast J \~N + \~N\ast \Phi l\tau l\Phi l
\~N = \Phi l\tau l\Phi
\ast
l .
This relation coincides with the same relation when x = l.
Simple test shows that N\ast
l
\r \Phi \ast \r \Phi Nl + \r K\ast \~\sigma \r K = \sigma l and \~N\r \Psi \r \Psi \ast \~N\ast + \r K\tau l\r K
\ast = \~\tau , therefore (151),
(152) are holding for the expansion
\circ
V2,
\circ +
V2 (38).
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TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 707
Theorem 7. Suppose that the simple isometric commutative expansion
\Bigl\{
Vs,
+
Vs
\Bigr\} 2
1
(14) corre-
sponding to the commutative system of operators T1 and T2 is such that:
1) \mathrm{d}\mathrm{i}\mathrm{m}E = \mathrm{d}\mathrm{i}\mathrm{m} \~E =< \infty , the operators \sigma 1, \~\sigma 1, N1, \~N1 are invertible, and the spectrum of the
operator T1 lies on the unit circle \BbbT ;
2) the characteristic function Sl(z) (19) of the operator T1 is invertible at least at one point
z \in \BbbC , the matrix-valued measure at and the function \varphi t from the multiplicative representation of
Sx(z) (19) are such that the operators Nx and \gamma x exist as the solutions of (21), (22), and there exist
\sigma x, \tau x, for which the relations (26) and (28) respectively are true.
Then the expansion
\Bigl\{
Vs,
+
Vs
\Bigr\} 2
1
is unitarily equivalent to the simple part of the isometric commu-
tative expansion
\Bigl\{ \circ
Vs,
\circ +
Vs
\Bigr\} 2
1
(38) of the commutative operator system (33) where \r \Psi and \r \Phi are defined
by the formulas (34) and (37) respectively.
Proof. To prove that the relations (153) – (155) are true for the expansions
\circ
Vs,
\circ +
Vs (38), we note
that equality (155), NK = K \~N, follows from the equality Nx\Phi
\ast
x = \Phi \ast
x
\~N in an obvious manner when
x = l since N = Nl, N0 = \~N and \r K = \Phi \ast
l . Moreover, relation (154) follows from the intertwining
condition (20) when x = l after proceeding to limit as z \rightarrow \infty . To prove the first of the relations in
(153), we consider
\r T \ast
2 \xi \Psi xe
i\varphi x - \r T \ast
1 \xi Nl\Psi xe
i\varphi x = \xi
\bigl\{
\Psi xJ
\bigl(
Nxe
i\varphi x + \gamma x
\bigr)
Jei\varphi x - Nl\Psi xe
2i\varphi x
\bigr\}
-
- 2\xi
l\int
x
\Psi tat\Psi
\ast
t e
- i\varphi sdt\Phi \ast - 1
l NxJe
i\varphi x + 2\xi
l\int
x
Nt\Psi tat\Phi
\ast
t e
- i\varphi sdt\Phi \ast - 1
l Jei\varphi x .
It is obvious that
\r T \ast
2 \xi \Psi xe
i\varphi x - \r T \ast
1 \xi Nl\Psi xe
i\varphi x = \xi Fxe
i\varphi x ,
where the function Fx is given by
Fx = \Psi xJ\gamma xJ - 2
l\int
x
\Psi tat\Phi
\ast
t e
i\varphi sdt\Phi \ast - 1
l NxJ + 2
l\int
x
Nt\Psi tat\Phi
\ast
t e
i\varphi sdt\Phi \ast - 1
l J,
since Nl\Psi x = \Psi xJNxJ. Elementary calculations using (211), (212), and (22) show that F \prime
x = FxaxJ.
Taking into account that Fl = \gamma lJ, we obtain that Fx = \gamma l\Psi x. This proves the necessity.
To prove the second of the relations in (153), T \ast
2\Psi
\ast - T \ast
1\Psi
\ast \~N\ast = \Psi \ast \~\gamma \ast , in view of (34), we
denote \r T \ast
2 \xi \Phi x - \r T \ast
1 \xi
\~N\ast \Phi x = \xi Gx where the operator-function Gx equals to
Gx = \Phi x\gamma
\ast
x - 2
x\int
0
\Phi te
- i\varphi satJN
\ast
t \Phi
- 1
t dt\Phi x + 2
x\int
0
\Phi tN
\ast
t e
- i\varphi satJ\Phi
- 1
t dt\Phi x.
As in the previous case, it is easy to verify that Gx satisfies the equation G\prime
x+GxaxJ = 0, and since
G0 = \~\gamma \ast , it is obvious that Gx = \~\gamma \ast \Phi x. So both of relations (153) are proved.
To use the theorem of the unitary equivalence [1], it is necessary to verify that the characteristic
function \r S(z) = \r K +\r \Psi
\bigl(
zI - \r T1
\bigr) - 1\r \Phi of the expansion \r V1 (38) coincides with Sl(z) (19). Consider
the vector-function f =
\bigl(
zI - \r T1
\bigr) - 1\r \Phi \xi which, obviously, is a solution of the equation
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
708 R. HATAMLEH, V. A. ZOLOTAREV
\bigl(
z - ei\varphi x
\bigr)
f(x) + 2
l\int
x
f(t)at\Phi
\ast
tdt\Phi
\ast - 1
x Jei\varphi x =
\surd
2\xi \Psi xe
i\varphi x . (40)
The function F (x) = (ze - i\varphi x - 1)f(x)J satisfies the following equation:
F (x) + 2
l\int
x
1
ze - i\varphi x - 1
F (t)Jat\Phi
\ast
tdt\Phi =
\surd
2\xi \Psi xJ.
It is obvious that F (x) satisfies the Cauchy problem
F \prime (x) =
z + e - i\varphi x
z - e - i\varphi x
F (x)Jax,
F (l) =
\surd
2\xi ,
(41)
whose solution is well-known [2] as
F (x) =
\surd
2\xi
\curvearrowleft
l\int
x
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
ei\varphi t + z
ei\varphi t - z
JdFt
\biggr\}
. (42)
Now it follows from (34) that
\r \Psi
\Bigl(
zI - \r T1
\Bigr) - 1
\r \Phi \xi = -
\surd
2
l\int
x
f(t)at\Phi
\ast
t dt,
then from equation (40), when x = 0, we have \r \Psi
\bigl(
zI - \r T1
\bigr) - 1\r \Phi \xi =
1\surd
2
F (0) - \xi \Psi 0J and since
\Psi 0J = \Phi \ast
l =
\r K we finally obtain \r S(z)\xi =
1\surd
2
F (0) and so \r S(z) = Sl(z) (19).
Theorem 7 is proved.
In the case when the spectrum of the operator T1 lies outside the unit circle \BbbT (this means that
in representation (4) there are no multiplier of the form (5)), it’s also possible to use the methods
presented above to construct the triangular model of the commutative operator system \{ T1, T2\} .
6. Since the simple component of the triangular model
\bigl\{
\r T1, \r T2
\bigr\}
(32) in L2
r,l(Fx) (6) is given by
[6, 7]
\r H1 = \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}
\Bigl\{
\r Rz
\r \Phi g + \r R\ast
w
\r \Psi \ast f : g, f \in Er; z, w \in \BbbC
\Bigr\}
, (43)
where z and w are points of regularity of the resolvent \r Rz =
\bigl(
zI - \r T1
\bigr)
of the operator \r T1, consider
the vector-function g(x, z) \in L2
r,l(Fx),
g(x, z) = \r Rz
\r \Phi g, (44)
where g \in Er and z /\in \sigma
\bigl(
\r T1
\bigr)
. Then it follows from the triangular model (32) that\Bigl\{
\r T2 - JNxJ\r T1 - J\gamma xJ
\Bigr\}
g(x, z) = 0. (45)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 709
Moreover, we suppose that the matrices-functions JNxJ and J\gamma xJ acting as multiplication from the
right by every function f(x) \in L2
r,l(Fx).
Denote by f(x, z) the following vector-function from L2
r,l(Fx):
f(x,w) = \r R\ast
w
\r \Psi \ast f, (46)
where f \in Er and w /\in \sigma
\bigl(
\r T1
\bigr)
. It is easy to see that
f(x.w) =
\surd
2
e - i\varphi x - \=w
fS\ast
x(w), (47)
where the operator-function Sx(w) is given by (19).
Now we calculate how the model operators \r T1 and \r T2 (32) act on f(x,w) (46). Since \r T \ast
1
\r R\ast
w =
= \=w\r R\ast
w - I, then
\r T \ast
1
\r R\ast
w
\r \Psi = \r T1
1
\=w
\Bigl\{
\r T \ast
1
\r R\ast
w + I
\Bigr\}
\r \Psi \ast =
1
\=w
\Bigl\{
\r R\ast
w
\r \Psi \ast - \r \Phi JS\ast
\Delta (w)
\Bigr\}
,
in view of \r T1
\r T \ast
1 +\r \Phi J\r \Phi \ast = I, \r T1
\r \Psi \ast +\r \Phi J \r K = 0. Therefore
\=w\r T1
\r R\ast
w
\r \Psi \ast = \r R\ast
w
\r \Psi \ast - \r \Phi JS\ast
\Delta (w). (48)
Now using the colligation relation (see (153), we obtain
\r T2
\r R\ast
w
\r \Psi \~N\ast =
1
\=w
T2
\Bigl\{
\r R\ast
w
\r T \ast
1
\r \Psi \ast \~N\ast + \r \Psi \~N\ast
\Bigr\}
=
=
1
\=w
T2
\r R\ast
w
\Bigl(
\r T \ast
2
\r \Psi \ast - \r \Psi \ast \~\gamma \ast
\Bigr)
- 1
\=w
\r \Phi \tau l\r K
\ast =
1
\=w
\r R\ast
w
\r \Psi \ast - 1
\=w
\r T2
\r R\ast
w
\r \Psi \ast \~\gamma \ast - 1
\=w
\r \Phi \tau lS
\ast
\Delta (w),
since \r T2
\r T \ast
2 +\r \Phi \tau l\r \Phi
\ast = I, \r T2
\r \Psi \ast \~N\ast +\r \Phi \tau l\r K
\ast = 0. Hence
\r T2
\r R\ast
w
\r \Psi \ast
\Bigl(
\~N\ast \=\Psi + \~\gamma \ast
\Bigr)
= \r R\ast
w
\r \Psi \ast - \r \Phi \tau lS
\ast
\Delta (w). (49)
Subtracting (49) from (48), we get
\=w\r T1
\r R\ast
w
\r \Psi \ast f - \r T2
\r R\ast
w
\r \Psi \ast
\Bigl(
\~N\ast \=\Psi + \~\gamma \ast
\Bigr)
f = \r \Phi (\tau 1 - J)S\ast
\Delta (w)f, (50)
where f \in Er. It follows from (46), (47) and the intertwining condition (20) that
\r R\ast
w
\r \Psi \ast
\Bigl(
\~N\ast \=\Psi + \~\gamma \ast
\Bigr)
f =
\surd
2
e - i\varphi x - \=w
f
\Bigl(
\~N\ast \=w + \~\gamma \ast
\Bigr)
S\ast
\Delta (w) = f(x,w) (N\ast
x \=w + \~\gamma \ast ) .
Therefore we can write (50) in the form
\=w\r T1f(x.w) -
\Bigl(
\~N\ast \=\Psi + \~\gamma \ast
\Bigr)
\r T2f(x,w) = \r \Phi (\tau l - J)S\ast
\Delta (w)f,
and in view of (46), (48) we finally obtain\Bigl\{
\r T1 - N\ast
x
\r T2 - \~\gamma \ast \r T1
\r T2
\Bigr\}
f(x,w) =
1
\=w
\Bigl[
\r \Phi (\tau l - J)S\ast
\Delta (w) + \~\gamma \ast \r T2
\r \Phi JS\ast
\Delta (w)f
\Bigr]
. (51)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
710 R. HATAMLEH, V. A. ZOLOTAREV
Let us show that the right-hand side of equality (51) belongs to the closed linear span generated by
the functions g(x, z) (44). From the equality \r T2
\r \Phi = \r T1
\r \Phi Nl + \r \Phi \gamma l (see (153)) it follows that it is
sufficient to prove that the functions
\~\gamma \ast \r T1
\r \Phi NlJS
\ast
\Delta (w)f, \~\gamma \ast \r \Phi \gamma lJS
\ast
\Delta (w)f,
have this property.
These functions belong to the subspace generated by g(x, z). The proofs are analogous. So, we
exhibit the proof of the second only.
Really,
\gamma x\r \Phi \gamma lJS
\ast
\Delta (w)f =
\surd
2fS\ast
\Delta (w)J\gamma l\Psi x\gamma
\ast
xe
i\varphi x =
\surd
2fS\ast
\Delta (w)J\gamma l\gamma
\ast
l \Psi xe
i\varphi x ,
in view of (37) and the fact that \Psi x\gamma
\ast
x = \gamma \ast l \Psi x.
Since the right-hand side of equality (51) belongs to the subspace generated by the functions
g(x, z) (44), then it follows from (45) that\Bigl[
\r T2 - JNxJ\r T1 - J\gamma xJ
\Bigr] \Bigl[
\r T1 - N\ast
x
\r T2 - \gamma \ast x
\r T1
\r T2
\Bigr]
f(x) = 0,
for all f(x) \in \r H1 (42). From this it easily follows that \BbbP x
\bigl(
\r T1, \r T2
\bigr)
f(x) = 0 where
\BbbP x (z1, z2) = \mathrm{d}\mathrm{e}\mathrm{t} \{ [Nxz1 - z2I + \gamma x] [z1 - N\ast
xz2 - \gamma \ast xz1z2]\} .
It follows from the intertwining relation (20) that
\mathrm{d}\mathrm{e}\mathrm{t} [Nxz1 - z2I + \gamma x] = \mathrm{d}\mathrm{e}\mathrm{t}
\Bigl[
\~Nxz1 - z2I + \~\gamma
\Bigr]
= \~\BbbQ (z1, z2),
\mathrm{d}\mathrm{e}\mathrm{t} [N\ast
xz2 - z1I + \gamma \ast xz1z2] = \mathrm{d}\mathrm{e}\mathrm{t}
\Bigl[
\~N\ast z2 - z1I + \~\gamma \ast z1z2
\Bigr]
= \~\BbbQ \ast (z1, z2) ,
so the polynomial \BbbP x (z1, z2) = \BbbP (z1, z2) = \~\BbbQ (z1, z2) \~\BbbQ \ast (z1, z2) do not depend on x. Thus, the
following theorem shows.
Theorem 8. Suppose that the simple commutative isometric expansion
\bigl\{
Vs,
+
Vs
\bigr\} 2
1
(14) corre-
sponding to the commutative operator system \{ T1, T2\} is such that the suppositions of Theorem 6 are
true. Then the T1, T2 operators annul the polynomial P (z1, z2),
\BbbP (T1, T2) = 0,
where P (z1, z2) = \~Q(z1, z2) \~Q
\ast (z1, z2).
This theorem represents an analogue of the Hamilton – Cayley theorem for the system of the
commuting operators \{ T1, T2\} close to the unitary ones and contains several fundamental distinctions
from the well-known result of M. S. Livšic [5] for the nonself-adjoint commutative operator systems.
Note that the polynomial \BbbP (z1, z2) has the following symmetry relatively to the unit circle \BbbT :
\BbbP
\biggl(
1
z1
,
1
z2
\biggr)
= (z1z2)
- 2r \BbbP (z1, z2) .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
TRIANGULAR MODELS OF COMMUTATIVE SYSTEMS OF LINEAR OPERATORS CLOSE . . . 711
References
1. Zolotarev V. A. Model representations of commutative systems of linear operators’ // Func. Analiz i Pril. – 1988. –
22, № 1. – P. 55 – 57.
2. Zolotarev V. A. Analytic methods of spectral representations of nonself-adjoint and nonunitary operators (in Russian). –
Kharkov: Kharkov Nat. Univ. Publ. House, 2003.
3. Brodskii M. S. Unitary operator colligations and their characteristic functions’ // Russ. Math. Surv. – 1978. – 33,
№ 4. – P. 159 – 192.
4. Zolotarev V. A. Time cones and a functional model on a Riemann suface // Math. USSR Sb. – 1991. – 70, № 2. –
P. 399 – 429.
5. Livšic M. S., Kravitsky N., Markus A., Vinnikov V. Theory of commuting nonself-adjoint operators. – Dordrecht, etc:
Kluwer Acad. Publ., 1995.
6. Zolotarev V. A. Isometric expansions of commutative systems of linear operators’ (in Russian) // Math. Phys., Anal.,
Geom. – 2004. – 11, № 3. – P. 282 – 301.
7. Zolotarev V. A. On isometric dilations of commutative systems of linear operators’ // J. Math. Phys., Anal., Geom. –
2005. – 1, № 2. – P. 192 – 208.
8. Lax P. Integrals of nonlinear equations of evolution and solitary waves’ // Communs Pure and Appl. Math. – 1968. –
21, № 5. – P. 467 – 490.
Received 07.08.14
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
|
| id | umjimathkievua-article-1872 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:19Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ba/978647b8c5abb883fc145d7595df43ba.pdf |
| spelling | umjimathkievua-article-18722019-12-05T09:30:15Z Triangular models of commutative systems of linear operators close to unitary operators Трикутнi моделi комутативних систем лiнiйних операторiв, близьких до унiтарних Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. Triangular models are constructed for commutative systems of linear bounded operators close to unitary operators. The construction of these models is based on the continuation of basic relations for the characteristic function along the general chain of invariant subspaces. Побудовано трикутнi моделi комутативних систем лiнiйних обмежених операторiв, близьких до унiтарних. Побудову цих моделей засновано на подовженнi основних спiввiдношень для характеристичної функцiї вздовж загального ланцюжка iнварiантних просторiв. Institute of Mathematics, NAS of Ukraine 2016-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1872 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 5 (2016); 694-711 Український математичний журнал; Том 68 № 5 (2016); 694-711 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1872/854 Copyright (c) 2016 Hatamleh R.; Zolotarev V. A. |
| spellingShingle | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. Triangular models of commutative systems of linear operators close to unitary operators |
| title | Triangular models of commutative systems of linear operators close to unitary operators |
| title_alt | Трикутнi моделi комутативних систем лiнiйних операторiв, близьких до унiтарних |
| title_full | Triangular models of commutative systems of linear operators close to unitary operators |
| title_fullStr | Triangular models of commutative systems of linear operators close to unitary operators |
| title_full_unstemmed | Triangular models of commutative systems of linear operators close to unitary operators |
| title_short | Triangular models of commutative systems of linear operators close to unitary operators |
| title_sort | triangular models of commutative systems of linear operators close to unitary operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1872 |
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