Jacobi operators and orthonormal matrix-valued polynomials. I
It is shown that every self-adjoint operator in a separable Hilbert space is unitarily equivalent to a block Jacobi operator. A system of orthogonal operator-valued polynomials is constructed.
Gespeichert in:
| Datum: | 2017 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2017
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1688 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507524390715392 |
|---|---|
| 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 | 2019-12-05T09:23:56Z |
| description | It is shown that every self-adjoint operator in a separable Hilbert space is unitarily equivalent to a block Jacobi operator.
A system of orthogonal operator-valued polynomials is constructed. |
| first_indexed | 2026-03-24T02:10:41Z |
| 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 and V. N. Karazin Kharkov Nat. Univ.)
JACOBI OPERATORS AND ORTHONORMAL
MATRIX-VALUED POLYNOMIALS. I
ОПЕРАТОРИ ЯКОБI ТА ОРТОГОНАЛЬНI
ОПЕРАТОРНОЗНАЧНI ПОЛIНОМИ. I
It is shown that every self-adjoint operator in a separable Hilbert space is unitarily equivalent to a block Jacobi operator.
A system of orthogonal operator-valued polynomials is constructed.
Показано, що будь-який самоспряжений оператор, заданий у сепарабельному гiльбертовому просторi, унiтарно
еквiвалентний блочному оператору Якобi. Побудовано систему ортогональних операторнозначних полiномiв.
Introduction. Jacobi matrix is the canonical form of a self-adjoint operator with simple spectrum [1],
spectral analysis of this matrix is tightly bound with the study of orthogonal polynomials [2, 3]. This
realm of analysis has deep connections with moment problem; interpolation problems; issues of the
extension of symmetrical operators, etc. [2, 3].
This work develops studies in this direction. At the beginning (Section 1), it is shown that every
self-adjoint operator acting in a separable Hilbert space is realized by the block Jacobi operator (is
unitarily equivalent to it), besides, sizes of the blocks correspond to the multiplicity of the spectrum of
the initial operator. Section 2 is devoted to the construction of the system of orthogonal matrix-valued
polynomials. These problems (in the matrix case) are studied in the works [4 – 8]. Important results
in spectral analysis obtained in [4, 5] found their fruitful application in the problem of moments.
Generalization of the scalar case [2] on the matrix-valued case is studied in [7, 8] and is represented
in the overall survey [6]. Establishment of links between block Jacobi matrices and theory of nonself-
adjoint operators with analytical analogues of the L. de Branges spaces of entire functions is the aim
of the present paper. The polynomials of the first and the second kind are constructed in Section 2
using an introduced notion of the measure nondegenerateness (see an analogue in [6]), and then the
operator-valued function with J -properties is constructed and its multiplicative expansion is obtained
using the methods of J -theory of V. P. Potapov [9, 10].
Constructions stated in this paper refer to the so called ”truncated” problem (n \in \BbbN ), i.e., the
finite block Jacobi matrix.
1. Block Jacobi operators. I. Consider the spectral resolution [1]
A =
\int
\BbbR
\lambda dE\lambda (1)
of a linear self-adjoint operator A given in a separable Hilbert space H. Let us select some vector
f1 from a dense in H set (in view of the separation property) and construct the subspace
L(f1)
df
= span\{ E\Delta f1 : \Delta \in \BbbR \} , (2)
c\bigcirc R. HATAMLEH, V. A. ZOLOTAREV, 2017
228 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. I 229
where \Delta runs over the totality of all the intervals of the axis \BbbR . Obviously, L1(f) and its orthogonal
complement H1 = H\ominus L (f1) are Et-invariant (t \in \BbbR ). Since H1 is also separable, then, selecting f2
from a countable dense set in H1, we define the subspace L (f2) (2) in H1. Repeating this procedure
of removal L (fk) countable number of times, we obtain
H =
\infty \sum
k=1
\oplus L(fk). (3)
Note that this procedure can terminate after finite number of steps and even on the first step. The latter
leads to the representation by the classical Jacobi matrix with scalar entries. Define the generating
subspace [1] G for the operator A,
G
df
= span\{ fk : k \in \BbbN \} , (4)
then (3) implies
H = span\{ E\Delta g : g \in G; \Delta \in \BbbR \} , (5)
where, as usual, \Delta belongs to the set of all intervals from \BbbR . Consider some Hilbert space E
(dimE \geq dimG), and let \psi be a linear bounded operator from E onto G. Define a nondecreasing
operator-function in E
F (x)
df
= \psi \ast Ex\psi , x \in \BbbR . (6)
As a \psi we can take, for example, the orthoprojector PG on G and suppose that E = G.
Denote by L2
\BbbR
\bigl(
E, dF (x)
\bigr)
the Hilbert space of E -valued vector-functions on \BbbR ,
L2
\BbbR
\bigl(
E, dF (x)
\bigr) df
=
\left\{ f(x) :
\int
\BbbR
\bigl\langle
dF (x)f(x), f(x)
\bigr\rangle
E
<\infty
\right\} (7)
which is generated as a result of the closure of linear span of finite continuous functions f(x) and
subsequent factorization by the kernel of metrics (7). This definition is correct (see [3, 11]). Specify
the linear operator U,
U : L2
\BbbR
\bigl(
E, dF (x)
\bigr)
\rightarrow H, f = Uf(x), f
df
=
\int
\BbbR
dEx\psi f(x). (8)
Image of the operator U is dense in H since vectors\int
\BbbR
dEx\psi \chi \Delta (x) = E\Delta \psi f, f \in E, \Delta \in \BbbR ,
linear span of which is dense in H, belong to it, in view of (5) (\psi f = g \in G).
If g(x) \in L2
\BbbR
\bigl(
E, dF (x)
\bigr)
, and f is given by (8), then
\bigl\langle
Etf, \psi g(x)
\bigr\rangle
=
t\int
- \infty
\bigl\langle
dF (s)f(s), g(x)
\bigr\rangle
. (9)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
230 R. HATAMLEH, V. A. ZOLOTAREV
Let f(x) from (8) be differentiable and f \prime (x) \in L2
\BbbR
\bigl(
E, dF (x)
\bigr)
, then
\| f\| 2 =
\int
\BbbR
\bigl\langle
f, dEx\psi f(x)
\bigr\rangle
=
\int
\BbbR
d
\bigl\langle
f,Ex\psi f(x)
\bigr\rangle
-
\int
\BbbR
\bigl\langle
Exf, \psi f
\prime (x)
\bigr\rangle
dx =
=
\int
\BbbR
d
x\int
- \infty
\langle dF (t)f(t), f(x)\rangle -
\int
\BbbR
x\int
- \infty
\bigl\langle
dF (t)f(t), f \prime (x)
\bigr\rangle
dx =
\int
\BbbR
\bigl\langle
dF (x)f(x), f(x)
\bigr\rangle
,
in view of (9). So, U (8) is isometrical on the dense set in L2
\BbbR
\bigl(
E, dF (x)
\bigr)
and thus the operator U
is unitary.
Let f be given by (8) where f(x) belongs to the linear span of continuous functions in
L2
\BbbR
\bigl(
E, dF (x)
\bigr)
. Then for all h \in H
\langle Af, h\rangle =
\int
\BbbR
x
\bigl\langle
f, dExh
\bigr\rangle
=
\int
\BbbR
x
\int
\BbbR
\bigl\langle
dEs\psi f(s), dExh
\bigr\rangle
=
=
\int
\BbbR
xdx
\int
\BbbR
\bigl\langle
dEs\psi f(s), Exh
\bigr\rangle
=
\int
\BbbR
xd
x\int
- \infty
\bigl\langle
dEs\psi f(s), h
\bigr\rangle
=
\int
E
\bigl\langle
dEx\psi xf(x), h
\bigr\rangle
,
consequently,
Af =
\int
\BbbR
dEx\psi (xf(x)), (10)
and thus AU = UQ, where Q is the operator of multiplication by the independent variable in
L2
\BbbR
\bigl(
E, dF (x)
\bigr)
,
(Qf)(x)
df
= xf(x) (f(x) \in L2
\BbbR
\bigl(
E, dF (x)
\bigr)
. (11)
Isometricity of U (8) and (10) implies
\| Af\| 2 =
\int
\BbbR
x2
\bigl\langle
dF (x)f(x), f(x)
\bigr\rangle
= \| Qf(x)\| 2.
So f(x) belongs to the domain \frakD Q of the operator Q (11) then and only then when f (8) belongs
to the domain \frakD A of the operator A.
Theorem 1. An arbitrary self-adjoint operator A acting in a Hilbert space H is unitarily
equivalent to the operator Q (11) in L2
\BbbR
\bigl(
E, dF (x)
\bigr)
(7), AU = UQ, where U is given by (8); Ex is
the resolution of identity of the operator A; F (x) is given by formula (6); and \psi is a linear bounded
operator from E on the generating subspace G (4).
II. Let A be bounded self-adjoint operator, then
An\psi g =
\int
\BbbR
xndEx\psi g
make sense for all g \in G and all n \in \BbbZ +. Show that the linear span of these vectors is dense in H.
If a vector f \in H is such that f \bot An\psi g (for all g \in G and all n \in \BbbZ +), then, using representation
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. I 231
(8) for f, we obtain
0 = \langle An\psi g, f\rangle =
\int
\BbbR
\bigl\langle
dF (x)xng, f(x)
\bigr\rangle
.
Therefore f(x) \bot Pn(x), where Pn(x) is an arbitrary E -valued polynomial of the degree n. Since
the set of such polynomials is dense in L2
\BbbR
\bigl(
E, dF (x)
\bigr)
(dF (x) has the dense support), then f(x) = 0,
and so f = 0.
Theorem 2. For every bounded self-adjoint operator A acting in a separable Hilbert space H,
H = span
\bigl\{
An\psi g : g \in E,n \in \BbbZ +
\bigr\}
. (12)
In the case of unboundedness of A see [1, 2].
Define the subspaces
Hn
df
= span
\bigl\{
Ak\psi g : g \in E; 0 \leq k \leq n
\bigr\}
, n \in \BbbZ +, (13)
which are ordered by inclusion, Hk \subseteq Hs as s > k, and let
Gn
df
= Hn \ominus Hn - 1, n \in \BbbZ +, (14)
where H - 1 = \{ 0\} and G0 = G (4). Then
H =
\infty \sum
k=0
\oplus Gk. (15)
If gk \in Gk, k \in \BbbZ +, then \langle Agk, gs\rangle = 0 as s > k + 1 since Agk \in Hk+1; and, similarly,
\langle Agk, gs\rangle = \langle gk, Ags\rangle as k > s+ 1 (Ags \in Hs+1). So,
\langle Agk, gs\rangle = 0 as s > k + 1 and s < k - 1, k, s \in \BbbZ +,
therefore the operator A has the three-diagonal block structure corresponding to expansion (15),
A =
\left[
\widetilde A0
\widetilde B0 0 0 \cdot \cdot \cdot \widetilde B\ast
0
\widetilde A1
\widetilde B1 0 \cdot \cdot \cdot
0 \widetilde B\ast
1
\widetilde A2
\widetilde B2 \cdot \cdot \cdot
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot
\right]
, (16)
where \widetilde Ak = PkAPk : Gk \rightarrow Gk, \widetilde Bk = PkAPk+1 : Gk+1 \rightarrow Gk (Pk is the orthoprojector on Gk
(14), k \in \BbbZ +). The definition of Gk (14) yields dimGk \leq dimG, k \in \BbbZ +. For dimGk = dimG
we specify unitary operators Vk : Gk \rightarrow G. If dimGk < dimG, we can define isometric operators
Vk : Gk \rightarrow G. Consider the set of operators in G: Ak = Vk \widetilde AkV
\ast
k , Bk = Vk \widetilde BkV
\ast
k+1, k \in \BbbZ +, and
define the block Jacobi operator
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
232 R. HATAMLEH, V. A. ZOLOTAREV
JG
df
=
\left[
A0 B0 0 0 \cdot \cdot \cdot
B\ast
0 A1 B1 0 \cdot \cdot \cdot
0 B\ast
1 A2 B2 \cdot \cdot \cdot
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot
\right]
, (17)
in the Hilbert space l2\BbbZ +
(G). Let V = diag[I, V1, . . .] be an isometric (by construction) operator from
H (15) into l2\BbbZ +
(G), then V A = JGV.
Theorem 3. An arbitrary bounded self-adjoint operator acting in a separable Hilbert space H
is isometrically equivalent to the Jacobi operator JG (17) in the space l2\BbbZ +
(G).
2. Matrix-valued orthogonal polynomials. III. Spectral analysis of Jacobi matrices (dimG =
= 1) is closely linked with properties of orthogonal polynomials [1 – 6]. We proceed to the construc-
tion of matrix-valued orthogonal polynomials.
Definition 1. A measure dF (x) is said to satisfy the nd-condition (non degenerata) if for every
E -valued polynomial of the finite degree Pn(x) =
\sum n
k=0
xkgk, gk \in E, 1 \leq k \leq n, n \in \BbbZ +, the
estimation \int
\BbbR
\bigl\langle
dF (x)Pn(x), Pn(x)
\bigr\rangle
> \delta n
n\sum
k=0
\| gk\| 2, (18)
is true, besides, the number \delta n does not depend on the vectors \{ gk\} n1 , and \delta n > 0 for all n \in \BbbZ +.
Note that nd-condition is per se equivalent to the nondegenerateness of nontrivial measure in [6].
Theorem 4. If a self-adjoint bounded operator A acting in a separable Hilbert space H is such
that the measure dF (x) satisfies the nd-condition, where F (x) is given by (6) and dimE = r <\infty ,
then the vector An\psi g does not belong to the space Hn - 1 (13) for any g \in E and n \in \BbbN .
Proof. Assuming the contrary, we suppose that there is such a vector g \in E that An\psi g \in Hn - 1
for some n \in \BbbN , then
An\psi g +
n - 1\sum
k=1
Ak\psi gk = 0, gk \in E, 0 \leq k \leq n - 1.
This implies \int
\BbbR
dEt\psi Pn(t) = 0,
where Pn(t) = tng+
\sum n - 1
k=0
tkgk is a E -valued polynomial. Applying Ex to this equality, we obtain
x\int
- \infty
dEt\psi Pn(t) = 0 \forall x \in \BbbR ,
and thus
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. I 233
x\int
- \infty
\bigl\langle
dEt\psi Pn(t), \psi f
\bigr\rangle
=
x\int
- \infty
\bigl\langle
dF (t)Pn(t), f
\bigr\rangle
= 0 \forall f \in E.
Consequently,
0 =
\int
\BbbR
\varphi (x)d
x\int
0
\bigl\langle
dF (t)Pn(t), f
\bigr\rangle
=
\int
\BbbR
\bigl\langle
dF (x)Pn(x), \varphi (x)f
\bigr\rangle
,
for an arbitrary scalar function \varphi (x): \BbbR \rightarrow \BbbC , therefore\int
\BbbR
\bigl\langle
dF (x)Pn(x), f(x)
\bigr\rangle
= 0,
where f(x) is any function of the form f(x) =
\sum m
k=0
\varphi k(x)fk, m \in \BbbZ +. Assuming that f(x) =
= Pn(x), we obtain \int
\BbbR
\bigl\langle
dF (x)Pn(x), Pn(x)
\bigr\rangle
= 0,
which is contrary to the nd-condition.
Theorem 4 is proved.
This theorem yields that the vector
f = An\psi g - PHn - 1A
n\psi g
is nonzero for all g \in E and all n \in \BbbN
\bigl(
PHn - 1 is the orthoprojector on Hn - 1 (13)
\bigr)
. It is obvious
that PHn - 1f = 0, consequently, f \bot Hs (\forall s, 0 \leq s \leq n - 1). Since
f = An\psi g +
n - 1\sum
k=0
Ak\psi gk, (19)
then to each g \in E (dimE = r < \infty ) there corresponds the set of vectors \{ gk\} n - 1
0 from E.
Formula (19) follows from (13) when dimE <\infty . This correspondence defines the linear operators
Nkg
df
= gk, 0 \leq k \leq n - 1. Since dimE <\infty , Nk are bounded for all k.
Write the vector f (19) in the form
f =
\int
\BbbR
dEt\psi \widetilde Pn(t)g,
where \widetilde Pn(t) = tn + tn - 1Nn - 1 + . . .+N0. Orthogonality f \bot Hs, 0 \leq s \leq n - 1 signifies that\int
\BbbR
\widetilde P \ast
s (t)dF (t)
\widetilde Pn(t) = 0, 0 \leq s \leq n - 1.
The operator
Dn =
\int
\BbbR
\widetilde Pn(t)dF (t) \widetilde Pn(t)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
234 R. HATAMLEH, V. A. ZOLOTAREV
is nonnegative and invertible since the nd-condition (18) implies
\bigm\| \bigm\| \bigm\| D 1
2
n g
\bigm\| \bigm\| \bigm\| 2 > \delta n\| g\| 2. Therefore the
polynomial Pn(t) = \widetilde Pn(t)D
- 1
2
n is normalized to unity,\int
\BbbR
P \ast
n(t)dF (t)Pn(t) = IE .
Theorem 5. If a bounded self-adjoint operator A is such that the measure dF (x) has the nd-
property (18) (F (x) is given by (6) and Ex is the resolution of identity of the operator A and
dimE = r < \infty ), then there exists the family of matrix-valued in E polynomials \{ Pn(x)\} \infty 0 , such
that \int
\BbbR
P \ast
k (x)dF (x)Pn(x) = \delta k,nIE , k, n \in \BbbZ +, (20)
besides, degPn(x) = n and the leading coefficient of Pn(x) is invertible (\forall n \in \BbbZ +).
The expansion [2 – 6]
xPn(x) = Pn+1(x)B
(n+1)
n + Pn(x)B
(n)
n + . . .+ P0(x)B
(0)
n\bigl(
B
(s)
n are linear bounded operators in E, 0 \leq s \leq n + 1
\bigr)
and (20) imply that B(s)
n = 0 for
0 \leq s \leq n - 2, besides,
B(n+1)
n =
\int
\BbbR
xP \ast
n+1(x)dF (x)Pn(x), B(n)
n =
\int
\BbbR
xP \ast
n(x)dF (x)Pn(x),
B(n - 1)
n =
\int
\BbbR
xP \ast
n - 1(x)dF (x)Pn(x);
and thus B(n)
n =
\bigl(
B
(n)
n
\bigr) \ast
, B
(n - 1)
n =
\bigl(
B
(n)
n - 1
\bigr) \ast
. So the totality
\bigl\{
Pn(x)
\bigr\} \infty
1
is the solution of the
finite-difference equation
xPn(x) = Pn+1(x)Bn + Pn(x)Cn + Pn - 1(x)B
\ast
n - 1, n \in \BbbZ +, (21)
where P - 1(x)
df
= 0, Bn = B
(n+1)
n , Cn = B
(n)
n , n \in \BbbZ +. Invertibility of the leading coefficients
of the polynomials Pn(x) implies invertibility of all operators Bn. Therefore polynomials of the
first kind Pn(x) are found as the solutions of (21) unambiguously if we take into account the initial
conditions
P0(x) = D0, P1(x) = D0(xI - C0)B
- 1
0 , (22)
where D0 =
\bigl(
F (\infty ) - F ( - \infty )
\bigr) - 1/2
is an invertible positive operator. Expression
Qn(x)
df
=
\int
\BbbR
dF (\xi )
Pn(\xi ) - Pn(x)
\xi - x
, n \in \BbbZ +, (23)
defines [2] operator-valued polynomials of the second kind degQn(x) = n - 1, besides, Qn(x)
also satisfy the finite-difference equation (21) and the initial data
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. I 235
Q0(x) = 0, Q1(x) = D - 1
0 B - 1
0 . (24)
Construct the Jacobi operator
JE
df
=
\left[
C0 B\ast
0 0 0 \cdot \cdot \cdot
B0 C1 B\ast
1 0 \cdot \cdot \cdot
0 B1 C2 B\ast
2 \cdot \cdot \cdot
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot
\right]
(25)
by the coefficients \{ Bn, Cn\} \infty 0 from (21), then the recurrence relations (21) formally imply
\BbbP (x)JE = x\BbbP (x), \BbbQ (x)JE = x\BbbQ (x), (26)
where \BbbP (x) =
\bigl[
P0(x), P1(x), . . .
\bigr]
, \BbbQ (x) =
\bigl[
Q0(x), Q1(x), . . .
\bigr]
.
Let Yn = Yn(\lambda ) and Zn = Zn(w) be the solutions of (21) corresponding to \lambda and w accordingly
(\lambda , w \in \BbbC ). The Green formula [2 – 4, 6]
(\lambda - \=w)
n\sum
k=m
YkZ
\ast
k = Yn+1BnZ
\ast
n - YnB
\ast
nZ
\ast
n+1 - YmBm - 1Z
\ast
m - 1 + Ym - 1B
\ast
m - 1Z
\ast
m (27)
is true for all n, m \in \BbbN . In particular, for m = 1, if Yk = Pk(\lambda ), Zk = Pk(w), then taking into
account (22) we obtain the Christoffel – Darboux formula [2 – 4, 6]
(\lambda - \=w)
n\sum
k=0
Pk(\lambda )P
\ast
k (w) = Pn+1(\lambda )BnP
\ast
n(w) - Pn(\lambda )B
\ast
nP
\ast
n+1(w) (28)
for all n \in \BbbZ +. Assuming in (27) m = 1, Yk = Pk(\lambda ), Zk = Qk(w) and using (22), (24), we obtain
the equality
(\lambda - \=w)
n\sum
k=0
Pk(\lambda )Q
\ast
k(w) = Pn+1(\lambda )BnQ
\ast
n(w) - Pn(\lambda )B
\ast
nQ
\ast
n+1(w) + IE \forall n \in \BbbZ +. (29)
Finally, relation
(\lambda - \=w)
n\sum
k=0
Qk(\lambda )Q
\ast
k(w) = Qn+1(\lambda )BnQ
\ast
n(w) - Qn(\lambda )B
\ast
nQ
\ast
n+1(w) (30)
follows from (27) as m = 1 and Yk = Qk(\lambda ), Zk = Qk(w).
Lemma 1. If Pn(\lambda ) and Qn(\lambda ) are the solutions of the finite-difference equation (21) which
satisfy the conditions (22), (24), then the Liouville – Ostrogradsky formula [2 – 4, 6]\bigl\{
P \ast
n(
\=\lambda )Qn+1(\lambda ) - Q\ast
n(
\=\lambda )Pn+1(\lambda )
\bigr\}
Bn = IE \forall n \in \BbbZ + (31)
is true, besides,
P \ast
n(
\=\lambda )Qn(\lambda ) - Q\ast
n(
\=\lambda )Pn(\lambda ) = 0 \forall n \in \BbbZ +. (32)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
236 R. HATAMLEH, V. A. ZOLOTAREV
Proof. We prove both equalities (31), (32) simultaneously, using induction by n. For n = 0,
the truth of (31), (32) follows from the initial data (22), (24). Let the statement be proved for all k
(0 \leq k \leq n). Show that this implies (31), (32) for k = n + 1. Using (21) and invertibility of Bn,
we obtain
B\ast
n
\bigl[
P \ast
n+1(
\=\lambda )Qn+1(\lambda ) - Q\ast
n+1(
\=\lambda )Pn+1(\lambda )
\bigr]
Bn =
=
\bigl[
(\lambda - Cn)P
\ast
n(
\=\lambda ) - Bn - 1P
\ast
n - 1(
\=\lambda )
\bigr] \bigl[
Qn(\lambda )(\lambda - Cn) - Qn - 1(\lambda )B
\ast
n - 1
\bigr]
-
-
\bigl[
(\lambda - Cn)Q
\ast
n(
\=\lambda ) - Bn - 1Q
\ast
n - 1(
\=\lambda )
\bigr] \bigl[
Pn(\lambda )(\lambda - Cn) - Pn - 1(\lambda )B
\ast
n - 1
\bigr]
=
= (\lambda - Cn)
\bigl[
P \ast
n(
\=\lambda )Qn - 1(\lambda ) - Q\ast
n(
\=\lambda )Pn - 1(\lambda )
\bigr]
B\ast
n - 1 -
- Bn - 1
\bigl[
P \ast
n - 1(
\=\lambda )Qn(\lambda ) - Q\ast
n - 1(
\=\lambda )Pn(\lambda )
\bigr]
(\lambda - Cn) = 0,
in view of the supposition of induction. Similarly,\bigl[
P \ast
n+1(
\=\lambda )Qn+2(\lambda ) - Q\ast
n+1(
\=\lambda )Pn+2(\lambda )
\bigr]
Bn+1 =
= P \ast
n+1(
\=\lambda )
\bigl[
Qn+1(\lambda )(\lambda - Cn+1) - Qn(\lambda )Bn
\bigr]
- Q\ast
n+1(
\=\lambda )
\bigl[
Pn+1(\lambda )(\lambda - Cn+1) - Pn(\lambda )Bn
\bigr]
=
=
\bigl\{
Q\ast
n+1(
\=\lambda )Pn(\lambda ) - P \ast
n+1(
\=\lambda )Qn(\lambda )
\bigr\}
B\ast
n = I,
which was to be proved.
IV. Using the polynomials Pn(\lambda ) and Qn(\lambda ), we construct the operator-function
Wn(\lambda )
df
=
\Biggl[
Pn(\lambda ) Pn+1(\lambda )Bn
Qn(\lambda ) Qn+1(\lambda )Bn
\Biggr]
, (33)
besides, degWn(\lambda ) = n, and define the involution J in E \oplus E,
J
df
=
\Biggl[
0 iIE
- iIE 0
\Biggr]
. (34)
From formulas (28) – (30) follows that
Wn(\lambda )JW
\ast
n(w) - J =
\lambda - \=w
i
n\sum
n=0
\Biggl[
Pk(\lambda ) 0
Qk(\lambda ) 0
\Biggr] \Biggl[
P \ast
k (w) Q\ast
k(w)
0 0
\Biggr]
, (35)
and so Wn(\lambda ) (33) has J -properties [6, 7],
Wn(\lambda )JW
\ast
n(\lambda ) - J =
\left\{
\geq 0, \lambda \in \BbbC +,
= 0, \lambda \in \BbbR ,
\leq 0, \lambda \in \BbbC - .
(36)
Equation (21) yields
Wn(\lambda ) =Wn - 1(\lambda )bn(\lambda ), (37)
where
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. I 237
bn(\lambda ) =
\Biggl[
0 B\ast - 1
n - 1
B - 1
n - 1 B - 1
n (\lambda I - Cn)
\Biggr]
, n \in \BbbZ +, (38)
reckoning that B - 1
df
= I. Therefore
Wn(\lambda ) = U0
\curvearrowright
n\prod
k=1
bk(\lambda ), (39)
where U0 is J -unitary,
U0
df
=
\Biggl[
0 P0(\lambda )
- P - 1
0 (\lambda ) 0
\Biggr]
,
and so we can regard that W - 1(\lambda ) = U0. “Primary” factors bk(\lambda ) (38) also have J -properties (36),
since
bk(\lambda )Jb
\ast
k(w) - J =
\lambda - \=w
i
\Biggl[
0 0
0 B - 1
k - 1
\bigl(
B - 1
k - 1
\bigr) \ast
\Biggr]
,
b\ast k(w)Jbk(\lambda ) - J =
\lambda - \=w
i
\Biggl[
0 0
0 IE
\Biggr]
for all k \in \BbbZ +. Thus factorization (39) is realized in the context of the class of operator-functions
satisfying the relations (36), besides, the factors bk(\lambda ) (38) are constructed by the elements of the
Jacobi matrix JE (25). Factorization, similar to (38), is obtained in [9] in somewhat different form.
(35) implies Wn(\lambda )JW
\ast
n
\bigl(
\=\lambda
\bigr)
J = IE\oplus E , therefore the operator
W - 1
n (\lambda )
df
= JW \ast
n
\bigl(
\=\lambda
\bigr)
J =
\Biggl[
B\ast
nQ
\ast
n+1
\bigl(
\=\lambda
\bigr)
- B\ast
nP
\ast
n+1
\bigl(
\=\lambda
\bigr)
- Q\ast
n
\bigl(
\=\lambda
\bigr)
Pn
\bigl(
\=\lambda
\bigr) \Biggr]
, n \in \BbbZ +, (40)
is the right inverse for Wn(\lambda ) (33). The fact that W - 1
n (\lambda ) (40) is also the left inverse for Wn(\lambda )
follows from (31), (32).
Observation 1. The relations (28) – (30) provide Wn(\lambda ) (33) with the J -properties (36) and also
secure the existence of the right inverse W - 1
n (\lambda ) (40), and the equalities (31), (32) are equivalent
to W - 1
n (\lambda )Wn(\lambda ) = I. So the relations (28) – (30) and (31), (32) for the polynomials Pn(\lambda ) and
Qn(\lambda ) have a natural interpretation in terms of the J -properties of the function Wn(\lambda ).
The operator-function
Sn(\lambda )
df
=Wn(\lambda )W
- 1
n (0) (41)
also has the J -properties (36) and
Sn(\lambda )JS
\ast
n(w) - J =
\lambda - \=w
i
n\sum
k=0
\Biggl[
Pk(\lambda ) 0
Qk(\lambda ) 0
\Biggr] \Biggl[
P \ast
k (w) Q\ast
k(w)
0 0
\Biggr]
, (42)
in virtue of the J -unitarity of W - 1
n (0). The function Sn(\lambda ) is such that Sn(0) = I. It is easy to
show that Sn(\lambda ) is equal
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
238 R. HATAMLEH, V. A. ZOLOTAREV
Sn(\lambda ) =
\Biggl[
An(\lambda ) Bn(\lambda )
Cn(\lambda ) Dn(\lambda )
\Biggr]
, (43)
where
An(\lambda )
df
= Pn(\lambda )B
\ast
nQ
\ast
n+1(0) - Pn+1(\lambda )BnQ
\ast
n(0) = I - \lambda
n\sum
k=0
Pk(\lambda )Q
\ast
k(0),
Bn(\lambda )
df
= Pn+1(\lambda )BnP
\ast
n(0) - Pn(\lambda )B
\ast
nP
\ast
n+1(0) = \lambda
n\sum
k=0
Pk(\lambda )P
\ast
k (0),
(44)
Cn(\lambda )
df
= Qn(\lambda )B
\ast
nQ
\ast
n+1(0) - Qn+1(\lambda )BnQ
\ast
n(0) = - \lambda
n\sum
k=0
Qk(\lambda )Q
\ast
k(0),
Dn(\lambda )
df
= Qn+1BnP
\ast
n(0) - Qn(\lambda )B
\ast
nP
\ast
n+1(0) = 1 + \lambda
n\sum
k=0
Qk(\lambda )P
\ast
k (0),
in virtue of the form of Wn(\lambda ) (33) and W - 1
n (0) (40), and also of (28) – (32). The functions (44) are
similar to the well-known scalar [2, 3] and matrix [7] functions, besides, degAn(\lambda ) = degBn(\lambda ) =
= n+ 1, degCn(\lambda ) = degDn(\lambda ) = n.
Observation 2. The functions (44), in spite of the properties following from (42), satisfy the
equalities
A\ast
n(
\=\lambda )Dn(\lambda ) - C\ast
n(
\=\lambda )Bn(\lambda ) = IE , D\ast
n(
\=\lambda )Bn(\lambda ) = B\ast
n(
\=\lambda )Dn(\lambda ),
A\ast
n(
\=\lambda )Cn(\lambda ) = C\ast
n(
\=\lambda )An(\lambda ), n \in \BbbZ +,
(45)
which are a corollary of the fact that JS\ast
n(
\=\lambda )J is the left inverse for Sn(\lambda ). The relations (45) can
be proved directly using (31), (32).
Observation 3. The normalization Et at zero for Sn(\lambda ) (41) is not binding. If we consider
Sn(\lambda , \lambda 0)
df
= Wn(\lambda )W
- 1
n (\lambda 0), \lambda 0 \in \BbbR , then: 1) Sn (\lambda 0, \lambda 0) = IE\oplus E ; 2) (42) take place in virtue of
the J -unitarity of W - 1
n (\lambda 0); 3) for Sn(\lambda , \lambda 0) representation (43) is true with the appropriate version
of the formulas (44).
(37), (41) imply
Sn(\lambda ) = Sn - 1(\lambda )an(\lambda ), n \in \BbbZ +, (46)
where the factor an(\lambda ) =Wn - 1(0)bn(\lambda )W
- 1
n (0) is equal
an(\lambda )
df
= I - i\lambda mnJ, n \in \BbbZ +, (47)
besides,
mn
df
=
\Biggl[
Pn(0)P
\ast
n(0) Pn(0)Q
\ast
n(0)
Qn(0)P
\ast
n(0) Qn(0)Q
\ast
n(0)
\Biggr]
\geq 0, n \in \BbbZ +, (48)
and mnJmn = 0, in virtue of (32). (46) implies
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. I 239
Sn(\lambda ) =
\curvearrowright
n\prod
k=0
ak(\lambda ), (49)
besides, S - 1(\lambda ) = I (W - 1(\lambda ) = U0). The factors ak(\lambda ) (47) have the J -properties because of
ak(\lambda )Ja
\ast
k(w) - J =
\lambda - \=w
i
mk, a\ast k(w)Jak(\lambda ) - J =
\lambda - \=w
i
JmkJ.
mkJmk = 0 yields that ak(\lambda ) (47) have the exponential representation
ak(\lambda ) = exp\{ - i\lambda mkJ\} , k \in \BbbZ +. (50)
Theorem 6. The operator-function Wn(\lambda ) (33) has the J -properties (36) and the multiplicative
expansion (39), where J and bk(\lambda ) are given by (34), (38).
The function Sn(\lambda ) (41) is expressed in terms of the functions (44) by formula (43), besides,
Sn(\lambda ) has the J -properties (36), and factorization (49) takes place, where the factors ak(\lambda ) are
given by (47), (50).
In the second part of this study the connection of these constructions with L. de Branges spaces
and nonself-adjoint operators will be established.
References
1. Akhiezer N. I., Glazman I. M. Theory of linear operators in Hilbert space. – 3rd ed. – Boston etc.: Pitman, 1981. –
Vols 1, 2.
2. Akhiezer N. I. The classical moment problem and some related questions in analysis. – Oliver \& Boyd, 1965.
3. Berezansky Yu. M. Expansion by eigenfunctions of self-adjoint operators (in Russian). – Kyiv: Naukova Dumka,
1965.
4. Arlinski\u i Yu., Klotz L. Weyl functions of bounded quasi-selfadjoint operators and block operator Jacobi matrices //
Acta Sci. Math. (Szeged). – 2010. – 70, № 3, 4. – P. 585 – 626.
5. Arlinski\u i Yu. Truncated Hamburger moment problem for an operator measure with compact support // Math. Nachr. –
2012. – 285, № 14, 15. – S. 1677 – 1695.
6. Damanik D., Pushnitskii A., Simon B. The analytical theory of matrix orthogonal polynomials // Sur. Approxim.
Theory. – 2008. – 4. – P. 1 – 85.
7. Lopez-Rodriguez P. The Nevanlinna parametrization for a matrix moment problem // Math. Scand. – 2001. – 89. –
P. 245 – 267.
8. Lopez-Rodriguez P. Riesz’s theorem for orthogonal matrix polynomials // Const. Approxim. – 1999. – 15, № 1. –
P. 135 – 151.
9. Potapov V. P. The multiplicative structure of J -contractive matrix functions (in Russian) // Tr. Mosk. Mat. Obshch. –
1955. – 4. – P. 125 – 236.
10. Zolotarev V. A. Analytic methods of spectral representations of non-selfadjoint and nonunitary operators (in Russian). –
Kharkov: KhNU Publ. House, 2003.
11. Malamud M. M., Malamud S. M. Spectral theory of operator measures in Hilbert space // St. Petersburg. Math. J. –
2004. – 15, № 3. – P. 323 – 373.
12. de Branges L. Hilbert spaces of entire functions. – London: Prentice-Hall, 1968.
13. Dyukarev Yu. M. Deficiency numbers of symmetric operators generated by block Jacobi matrices // Sb. Math. –
2006. – 197, № 8. – P. 1177 – 1204.
14. Woracek H. De Branges spaces and growth aspects // Operator Theory. – Basel: Springer, 2015.
15. Romanov R. Jacobi matrices and de Branges spaces // Operator Theory. – Basel: Springer, 2014.
Received 02.10.15,
after revision — 10.03.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
|
| id | umjimathkievua-article-1688 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:41Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5b/22eca24709b7a5215e95446d7a10115b.pdf |
| spelling | umjimathkievua-article-16882019-12-05T09:23:56Z Jacobi operators and orthonormal matrix-valued polynomials. I Оператори якобi та ортогональнi операторнозначнi полiноми. I Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. It is shown that every self-adjoint operator in a separable Hilbert space is unitarily equivalent to a block Jacobi operator. A system of orthogonal operator-valued polynomials is constructed. Показано, що будь-який самоспряжений оператор, заданий у сепарабельному гiльбертовому просторi, унiтарно еквiвалентний блочному оператору Якобi. Побудовано систему ортогональних операторнозначних полiномiв. Institute of Mathematics, NAS of Ukraine 2017-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1688 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 2 (2017); 228-239 Український математичний журнал; Том 69 № 2 (2017); 228-239 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1688/670 Copyright (c) 2017 Hatamleh R.; Zolotarev V. A. |
| spellingShingle | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. Jacobi operators and orthonormal matrix-valued polynomials. I |
| title | Jacobi operators and orthonormal matrix-valued polynomials. I |
| title_alt | Оператори якобi та ортогональнi
операторнозначнi полiноми. I |
| title_full | Jacobi operators and orthonormal matrix-valued polynomials. I |
| title_fullStr | Jacobi operators and orthonormal matrix-valued polynomials. I |
| title_full_unstemmed | Jacobi operators and orthonormal matrix-valued polynomials. I |
| title_short | Jacobi operators and orthonormal matrix-valued polynomials. I |
| title_sort | jacobi operators and orthonormal matrix-valued polynomials. i |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1688 |
| work_keys_str_mv | AT hatamlehr jacobioperatorsandorthonormalmatrixvaluedpolynomialsi AT zolotarevva jacobioperatorsandorthonormalmatrixvaluedpolynomialsi AT hatamlehr jacobioperatorsandorthonormalmatrixvaluedpolynomialsi AT zolotarʹovva jacobioperatorsandorthonormalmatrixvaluedpolynomialsi AT hatamlehr operatoriâkobitaortogonalʹnioperatornoznačnipolinomii AT zolotarevva operatoriâkobitaortogonalʹnioperatornoznačnipolinomii AT hatamlehr operatoriâkobitaortogonalʹnioperatornoznačnipolinomii AT zolotarʹovva operatoriâkobitaortogonalʹnioperatornoznačnipolinomii |