Jacobi operators and orthonormal matrix-valued polynomials. II
We use a system of operator-valued orthogonal polynomials to construct analogs of L. de Branges spaces and establish their relationship with the theory of nonself-adjoint operators.
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| author | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. |
| author_facet | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. |
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| description | We use a system of operator-valued orthogonal polynomials to construct analogs of L. de Branges spaces and establish
their relationship with the theory of nonself-adjoint operators. |
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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 Kharkiv Nat. Univ.)
JACOBI OPERATORS AND ORTHONORMAL
MATRIX-VALUED POLYNOMIALS. II
ОПЕРАТОРИ ЯКОБI ТА ОРТОГОНАЛЬНI
ОПЕРАТОРНОЗНАЧНI ПОЛIНОМИ. II
We use a system of operator-valued orthogonal polynomials to construct analogs of L. de Branges spaces and establish
their relationship with the theory of nonself-adjoint operators.
Iз використанням системи операторнозначних ортогональних полiномiв побудовано аналоги просторiв Л. де Бранжа
та встановлено їх зв’язок з теорiєю несамоспряжених операторiв.
1. Introduction. This work is based on the main constructions of the first part of the study
(Jacobi operators and orthonormal matrix-valued polynomials. I). Establishment of links between
the constructions of the first part (Jacobi operators and orthogonal polynomials) with the theory of
nonself-adjoint operators and further realization of these constructions in analogues of vector-valued
L. de Branges spaces is the aim of the present study. The works [14, 15] are dedicated to the
connection of Jacobi matrices with L. de Branges spaces. Method of reconstruction of the system
of orthogonal polynomials by the functions specifying the L. de Branges space is one of results of
this paper. In Section 1 using a system of orthonormal matrix-valued polynomials a Hilbert L. de
Branges space is constructed which is an analogue of the well-known L. de Branges space [10, 12]
in the discrete case. In Section 2 connection of these L. de Branges spaces with triangular models of
nonself-adjoint nilpotent operators [10] is established. In Section 3 resolvent of the truncated Jacobi
operator is calculated, and in the case of the finite dimensionality of E, an orthonormal basis of
generating kernels is found.
2. L. de Branges spaces. I. In this section we remind main definitions and facts from the first
part of the work. Denote by L2
\BbbR (E, dF (x)) the Hilbert space of E -valued vector-functions on \BbbR
(\mathrm{d}\mathrm{i}\mathrm{m}E = r < \infty ),
L2
\BbbR (E, dF (x)))
df
=
\left\{ f(x) :
\int
\BbbR
\langle dF (x)f(x), f(x)\rangle E < \infty
\right\} . (1)
Let the measure dF (x) satisfy the nd-condition\int
\BbbR
\langle dF (x)Pn(x), Pn(x)\rangle > \delta n
n\sum
k=0
\| gk\| 2 ,
for all Pn(x) =
\sum n
k=0
xkgk (gk \in E, 1 \leq k \leq n, n \in \BbbZ +). Then (see Part I) there exists the
family of matrix-valued polynomials \{ Pn(x)\} \infty 0 , such that\int
\BbbR
P \ast
k (x)dF (x)Pn(x) = \delta k,nIE , k, n \in \BbbZ +. (2)
c\bigcirc R. HATAMLEH, V. A. ZOLOTAREV, 2017
836 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. II 837
The polynomials \{ Pn(x)\} \infty 0 are the solutions of the finite-difference equations
xPn(x) = Pn+1(x)Bn + Pn(x)Cn + Pn - 1(x)B
\ast
n - 1, n \in \BbbZ +.
Let Qn(x) be polynomials of the second kind,
Qn(x)
df
=
\int
\BbbR
dF (\xi )
Pn(\xi ) - Pn(x)
\xi - x
, n \in \BbbZ +.
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] (3)
by \{ Bn, Cn\} \infty 0 . Define the operator-function
Wn(\lambda )
df
=
\biggl[
Pn(\lambda ) Pn+1(\lambda )Bn
Qn(\lambda ) Qn+1(\lambda )Bn
\biggr]
and the involution
J
df
=
\biggl[
0 iIE
- iIE 0
\biggr]
. (4)
Specify the operator-function
Sn(\lambda )
df
= Wn(\lambda )W
- 1
n (0) =
\biggl[
An(\lambda ) Bn(\lambda )
Cn(\lambda ) Dn(\lambda )
\biggr]
, (5)
besides,
An(\lambda ) = I - \lambda
n\sum
k=0
Pk(\lambda )Q
\ast
k(0),
Bn(\lambda ) = \lambda
n\sum
k=0
Pk(\lambda )P
\ast
k (0),
Cn(\lambda ) = - \lambda
n\sum
k=0
Qk(\lambda )Q
\ast
k(0),
Dn(\lambda ) = 1 + \lambda
n\sum
k=0
Q\ast
k(\lambda )P
\ast
k (0).
Then
Sn(\lambda ) = Sn - 1(\lambda )an(\lambda ), n \in \BbbZ +, (6)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
838 R. HATAMLEH, V. A. ZOLOTAREV
where
an(\lambda )
df
= I - i\lambda mnJ, 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 +. (7)
II. Consider the subspaces
\scrL n
df
= \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}
\Biggl\{
n\sum
k=0
Pk(x)fk : fk \in E(1 \leq k \leq n)
\Biggr\}
, n \in \BbbZ +, (8)
in L2
\BbbR (E, dF (x)) (1). The kernel
Kn(\lambda ,w)
df
=
n\sum
k=1
Pk(\lambda )P
\ast
k (w), \lambda , w \in \BbbC , (9)
is Hermitian positive [2, 3] and is generating in \scrL n,
f(\lambda ) =
\int
\BbbR
Kn(\lambda , x)dF (x)f(x)
for all f(\lambda ) \in \scrL n (\forall \lambda \in \BbbC ). The orthoprojector in L2
\BbbR (E, dF (x)) on \scrL n is given by
(P\scrL nf) (x)
df
=
\int
\BbbR
Kn(x, y)dF (y)f(y).
Following [2, 3], we every vector f = \mathrm{c}\mathrm{o}\mathrm{l} [f0, f1, . . .] , fk \in E, k \in \BbbZ +, from l2\BbbZ +
(E) juxtapose
with function f(x) \in L2
\BbbR (E, dF (x)),
f(x) = V f, f(x)
df
=
\infty \sum
k=0
Pk(x)fk, (10)
where \{ Pk(x)\} \infty 0 is a family of orthonormal (2) polynomials. Series (10) converges in the topology
of L2
\BbbR (E, dF (x)) for all f \in l2\BbbZ +
(E). Operator V (10) isometrically maps l2\BbbZ +
(E) on the subspace
in L2
\BbbR (E, dF (x)),
\scrL \infty
df
= \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n} \{ f(x) : f(x) \in \scrL n, n \in \BbbZ +\} ,
which is the closure of the linear span of the subspaces \scrL n (8). To calculate the inverse of V (10) is
to calculate the Fourier coefficients
fk =
\int
\BbbR
P \ast
k (x)dF (x)f(x), k \in \BbbZ +.
The operator V (10) is a unitary operator from l2\BbbZ +(n)(E) on \scrL n (8),
l2\BbbZ +(n)(E)
df
= \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}
\Biggl\{
f = \mathrm{c}\mathrm{o}\mathrm{l} [f0, f1, . . . , fn] : fk \in E;
n\sum
k=1
\| fk\| 2 < \infty
\Biggr\}
, (11)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. II 839
besides, \BbbZ +(n) = \{ k \in \BbbZ +; 0 \leq k \leq n\} , n \in \BbbZ +.
Let Ln(\lambda ) = [IE , 0]Sn(\lambda ),
Ln(\lambda ) = [An(\lambda ), Bn(\lambda )] , n \in \BbbZ +,
in view of (5). It is obvious that
Ln(\lambda )JL
\ast
n(w) =
\lambda - \=w
i
Kn(\lambda ,w),
besides, Kn(\lambda ,w) is given by formula (9). Consider the orthoprojectors P\pm =
1
2
(IE\oplus E \pm J) in
E \oplus E, then
Ln(\lambda )P+ =
1
2
En(\lambda ) [IE , iIE ] , Ln(\lambda )P - =
1
2
\widetilde En(\lambda ) [IE , - iIE ] ,
where the operator-functions En(\lambda ) and \widetilde En(\lambda ) in E equal
En(\lambda )
df
= An(\lambda ) - iBn(\lambda ), \widetilde En(\lambda )
df
= An(\lambda ) + iBn(\lambda ), \lambda \in \BbbC , n \in \BbbZ +. (12)
Using Ln(\lambda )JL
\ast
n(w) = Ln(\lambda )P+L
\ast
n(w) - Ln(\lambda )P - L
\ast
n(w), we obtain
Kn(\lambda ,w) = i
En(\lambda )E
\ast
n(w) - \widetilde En(\lambda ) \widetilde E\ast
n(w)
\lambda - \=w
, n \in \BbbZ +. (13)
Theorem 1. For all n \in \BbbZ + the operator-functions En(\lambda ), \widetilde En(\lambda ) (12) satisfy the relations
En(\lambda )E
\ast
n(\lambda ) - \widetilde En(\lambda ) \widetilde En(\lambda )
\left\{
> 0, \lambda \in \BbbC +,
= 0, \lambda \in \BbbR ,
< 0, \lambda \in \BbbC - ,
(14)
besides, the function En(\lambda ) ( \widetilde En(\lambda )) is invertible and \BbbC + (correspondingly, in \BbbC - ).
Proof. The relations (13) follow from the Hermitian positiveness of the kernel Kn(\lambda ,w) (12).
The inequality En(\lambda )E
\ast
n(\lambda ) - \widetilde En(\lambda ) \widetilde E\ast
n(\lambda ) > 0 for all \lambda \in \BbbC + follows from
En(\lambda )E
\ast
n(\lambda ) - \widetilde En(\lambda ) \widetilde E\ast
n(\lambda ) =
\lambda - \=\lambda
i
n\sum
k=0
Pk(\lambda )P
\ast
k (\lambda ) \geq
\lambda - \=\lambda
i
P0(\lambda )P
\ast
0 (\lambda ) =
=
\lambda - \=\lambda
i
D2
0 > 0,
in view of the invertibility of D. This implies the invertibility of E\ast
n(\lambda ) (and so the invertibility of
En(\lambda ) also) for all \lambda \in \BbbC +.
If for some x \in \BbbR there is such a subsequence \{ fs\} \infty 0 (\| fs\| = 1 \forall s \in \BbbZ +) that E\ast
n(x)fs \rightarrow 0,
s \rightarrow \infty , then Kn(\lambda , x)fs \rightarrow 0, s \rightarrow \infty , in virtue of (12), for all \lambda \in \BbbC . Thus\sum n
k=0
Pk(\lambda )P
\ast
k (x)fs \rightarrow 0, s \rightarrow \infty , therefore
\sum n
k=0
\| P \ast
k (x)fs\|
2 \rightarrow 0, s \rightarrow \infty , which is im-
possible, since \| P \ast
0 (x)fs\|
2 = \| D0fs\| 2 > 0.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
840 R. HATAMLEH, V. A. ZOLOTAREV
In the case of the self-adjointness of An(\lambda ) and Bn(\lambda ) as \lambda \in \BbbR , the functions \widetilde En(\lambda ) = E\ast
n
\bigl(
\=\lambda
\bigr)
and En(\lambda ) belong to the Hermite – Biehler class [14, 15].
The operator-function Lk(\lambda ) satisfies the recurrent relation
Lk(\lambda ) = Lk - 1(\lambda )ak(\lambda ), k \in \BbbZ +, (15)
in view of (6). Taking into account the form of ak(\lambda ) (7), we obtain that Lk(\lambda ) is the solution of
the system of equations
Lk(\lambda ) + i\lambda
k\sum
s=0
Ls - 1(\lambda )msJ = [IE , 0] , 0 \leq k \leq n, n \in \BbbZ +,
where mk are given by (7) and L - 1(\lambda )
df
= [IE , 0] . The J -properties of ak(\lambda ) (7) imply
Sn(\lambda )JS
\ast
n(w) - J =
\lambda - \=w
i
n\sum
k=0
Sk - 1(\lambda )mkS
\ast
k - 1(w),
and thus
Ln(\lambda )JL
\ast
n(w) =
\lambda - \=w
i
n\sum
k=0
Lk - 1(\lambda )mkL
\ast
k - 1(w),
so,
Kn(\lambda ,w) =
n\sum
k=0
Lk - 1(\lambda )mkL
\ast
k - 1(w). (16)
Define the weight spaces
l2\BbbZ +(n)(E \oplus E,m)
df
= \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}
\Biggl\{
f = [f0, . . . , fn] :
n\sum
k=0
\langle mkfk, fk\rangle < \infty
\Biggr\}
, (17)
where mk are given by (7); fk = \mathrm{c}\mathrm{o}\mathrm{l} [uk, vk] ; uk, vk \in E, 0 \leq k \leq n. Factorization by the
metric kernel is executed in this space (17). The spaces (17) are ordered by inclusion l2\BbbZ +(n)(E \oplus
\oplus E,m) \subseteq l2\BbbZ +(p)(E \oplus E,m) as n < p. The inner product in l\BbbZ +(n)(E \oplus E,m) is given by
\langle f, g\rangle =
\sum n
k=0
\langle mkfk, gk\rangle , where f = [f0, . . . , fn] , g = [g0, . . . , gn] are from l\BbbZ +(n)(E \oplus E,m).
III. Every f \in l2\BbbZ +(n)(E \oplus E,m) we juxtapose with the E -valued vector-function F (\lambda ),
F (\lambda ) = \scrB f df
=
n\sum
k=0
Lk - 1(\lambda )mkfk, (18)
assuming that Lk(\lambda ) is the solution of (15). The map \frakB is said to be the L. de Branges transform
[10, 12]. Obviously, \mathrm{d}\mathrm{e}\mathrm{g}F (\lambda ) \leq n. Describe the class of functions F (\lambda ) (18).
Lemma 1. For all n \in \BbbZ + the operator-functions En(\lambda ), \widetilde En(\lambda ) (12) are polynomials of degree
n+ 1, leading coefficients of which are invertible operators.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. II 841
Consider the expression E - 1
n (\lambda )F (\lambda ) where F (\lambda ) is given by (18) and \lambda \in \BbbC +. Theorem 1 yields
that this function is holomorphic in \BbbC + and E - 1
n (\lambda )F (\lambda ) \rightarrow 0 as \lambda \rightarrow \infty (since \mathrm{d}\mathrm{e}\mathrm{g}F (\lambda ) \leq n). In
connection with the fact that the function E - 1
n (x)F (x), x \in \BbbR , behaves at infinity as x - pN (\alpha \in \BbbN ,
N is a linear bounded operator in E ), the integral\int
\BbbR
\bigm\| \bigm\| E - 1
n (x)F (x)
\bigm\| \bigm\| 2
E
dx < \infty
converges. Similar fact takes place for the function \widetilde E - 1(\lambda )F (\lambda ) in \BbbC , besides,\int
\BbbR
\bigm\| \bigm\| E - 1
n (x)F (x)
\bigm\| \bigm\| 2
E
dx =
\int
\BbbR
\bigm\| \bigm\| \bigm\| \widetilde E - 1
n (x)F (x)
\bigm\| \bigm\| \bigm\| 2
E
dx,
since (E\ast
n(x))
- 1E - 1
n (x) =
\Bigl( \widetilde E\ast
n(x)
\Bigr) - 1 \widetilde E - 1
n (x) as x \in \BbbR in virtue of (14). As a result, we obtain
the inner description of the space of the function F (\lambda ) (17).
Definition 1. Let En(\lambda ) and \widetilde En(\lambda ) be operator-valued polynomials in E of degree n+ 1, the
leading coefficients of which are invertible operators, En(0) = \widetilde En(0) = I, and (14) take place,
besides, En(\lambda ) and \widetilde En(\lambda ) are invertible in the semiplanes \BbbC + and \BbbC - correspondingly.
The linear span of the entire E -valued (\mathrm{d}\mathrm{i}\mathrm{m}E < \infty ) functions F (\lambda ) is said to be the L. de
Branges space \scrB (An, Bn) (here 2An = En + \widetilde En, 2iBn = \widetilde En - En) if
a) E - 1
n (\lambda )F (\lambda ) ( \widetilde E - 1
n (\lambda )F (\lambda )) is holomorphic in \BbbC + (in \BbbC - ) function, besides, E - 1
n (\lambda )F (\lambda ) \rightarrow
\rightarrow 0 ( \widetilde E - 1
n (\lambda )F (\lambda ) \rightarrow 0) as \lambda \rightarrow \infty and \lambda \in \BbbC + (correspondingly, \lambda \in \BbbC - );
b) the integral \int
\BbbR
\bigm\| \bigm\| E - 1
n (x)F (x)
\bigm\| \bigm\| 2
E
dx =
\int
\BbbR
\bigm\| \bigm\| \bigm\| \widetilde E - 1
n (x)F (x)
\bigm\| \bigm\| \bigm\| 2
E
dx < \infty (19)
is finite.
The inner product in \scrB (An, Bn) is given by
\langle F (\lambda ), G(\lambda )\rangle = 1
2\pi i
\int
\BbbR
\bigl\langle
E - 1
n (x)F (x), E - 1
n (x)G(x)
\bigr\rangle
E
dx.
Factorization by the metric kernel is executed in the space \scrB (An, Bn) .
This definition of the L. de Branges space is the generalization of the classical L. de Branges
space [10, 12] and for vector-valued functions is new.
The function Kn(\lambda ,w) (13) is the reproducing kernel in \scrB (An, Bn) ,
\langle F (\lambda ),Kn(\lambda ,w)g\rangle \scrB = \langle F (w), g\rangle E (20)
for all F (\lambda ) \in \scrB (An, Bn) , all w \in \BbbC , and all g \in E.
Theorem 2. The operator \scrB (18) specifies the one-to-one correspondence between the spaces
l2\BbbZ +(n)(E \oplus E,m) (17) and \scrB (An, Bn) , besides, the Parseval equality
\langle F (\lambda ), G(\lambda )\rangle \scrB = \langle f, g\rangle l2 ,
takes place, where F (\lambda ) = \scrB f, G(\lambda ) = \scrB g, and f, g \in l2\BbbZ +(n)(E \oplus E,m).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
842 R. HATAMLEH, V. A. ZOLOTAREV
Proof. Since
\langle Kn(\lambda ,w)f, g\rangle E = \langle Kn(x,w)f,Kn(x, \lambda )g\rangle \scrB ,
where f, g \in E, then using (16) we obtain
\langle Kn(\lambda ,w)f, g\rangle E = \langle f(w), g(\lambda )\rangle l2 ,
besides, f(w) and g(\lambda ) are given by f(w) = [f0(w), . . . , fn(w)] , g(\lambda ) = [g0(\lambda ), . . . , gn(\lambda )]
(fk(w) = \mathrm{c}\mathrm{o}\mathrm{l}
\bigl[
A\ast
k - 1(w)f,B
\ast
k - 1(w)f
\bigr]
, gk(\lambda ) = \mathrm{c}\mathrm{o}\mathrm{l}
\bigl[
A\ast
k - 1(\lambda )g,B
\ast
k - 1(\lambda )g
\bigr]
, 0 \leq k \leq n) and
belong to space (17), for all \lambda , w \in \BbbC and all f, g \in E. Since \scrB f(w) = Kn(\lambda ,w)f and
\scrB g(w) = Kn(\lambda ,w)g, we obtain the Parseval equality (20).
To conclude the proof, we need to ascertain that Ln
df
= \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ f(w) : f \in E,w \in \BbbC \} coincides with
L2
\BbbZ +(n)(E \oplus E,m), and, secondly, to show that the space \scrB n
df
= \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n} \{ Kn(\lambda ,w)f : f \in E,w \in \BbbC \}
coincides with \scrB (An, Bn) . The equality \scrB n = \scrB (An, Bn) follows from (19); since if there is
the function F (\lambda ) \in \scrB (An, Bn) orthogonal to \scrB n, then, taking into account (20), we obtain
\langle F (w), f\rangle E = 0 for all w \in \BbbC and all f \in E, and thus F (\lambda ) \equiv 0.
Let Ln \not = l2\BbbZ +(n)(E \oplus E,m), then there is such a vector g \in L2
\BbbZ +(n)(E \oplus E,m) that g \bot Ln,
\langle f(w), g\rangle l2 =
\Biggl\langle
f,
n\sum
k=0
Lk - 1(w)mkgk
\Biggr\rangle
E
(\forall f \in E),
therefore
n\sum
k=0
Lk - 1(w)mkgk = 0,
Lk - 1(w)mk = [Pk(w)P
\ast
k (0), Pk(w)Q
\ast
k(0)] , k \in \BbbZ +,
therefore
0 =
n\sum
k=0
Lk - 1(w)mkgk =
n\sum
k=0
Pk(w) \{ P \ast
k (0)uk +Q\ast
k(0)vk\} ,
where gk = \mathrm{c}\mathrm{o}\mathrm{l} [uk, vk] , 1 \leq k \leq n. Taking into account the orthonormality (2) of the polynomials
\{ Pk(\lambda )\} n0 , we obtain
P \ast
k (0)uk +Q\ast
k(0)vk = 0, 0 \leq k \leq n.
This signifies that the norm of the vector g in L2
\BbbZ +(n)(E \oplus E,m) is equal to zero. So, g = 0.
3. Nonself-adjoint operators. IV. Specify the nonself-adjoint operator
(Anf)k
df
= - i
n\sum
s=k+1
Jmsfs, 0 \leq k \leq n, (21)
in l2\BbbZ +(n)(E \oplus E,m) (17), where J and mk are given by (4) and (7) (\mathrm{d}\mathrm{i}\mathrm{m}E < \infty ). It is obvious
that An (21) is nilpotent. Specifying the operator \varphi : l2\BbbZ +(n)(E \oplus E,m) \rightarrow E \oplus E by the formula
\varphi nf
df
=
n\sum
s=0
msfs
we obtain the colligation [10]
\Delta n =
\Bigl(
An, l
2
\BbbZ +(n)(E \oplus E,m), \varphi n, E \oplus E, - J
\Bigr)
. (22)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. II 843
Theorem 3. The characteristic Livsic – Brodskii function S\Delta n(\lambda ) = I - i\varphi (An - \lambda I) - 1 \varphi \ast
n( - J)
of the colligation \Delta n (22) coincides with Sn
\bigl(
\lambda - 1
\bigr)
(5), S\Delta n(\lambda ) = Sn
\bigl(
\lambda - 1
\bigr)
.
The L. de Branges transform \scrB (18) transforms the operator An (21) into the shift operator \widetilde An,\Bigl( \widetilde AnF
\Bigr)
(\lambda )
df
=
1
\lambda
(F (\lambda ) - F (0)), F (\lambda ) \in \scrB (An, Bn) . (23)
The operator \widetilde \varphi \ast
n = \scrB \varphi \ast
n equals
\widetilde \varphi \ast
ng
df
= e1(\lambda )u+ e2(\lambda )v, g = \mathrm{c}\mathrm{o}\mathrm{l}[u, v],
where
e1(\lambda ) = \lambda - 1Bn(\lambda ), e2(\lambda ) = \lambda - 1 (I - An(\lambda )) . (24)
So, we have the colligation
\widetilde \Delta n =
\Bigl( \widetilde An,\scrB (An, Bn) , \widetilde \varphi n, E \oplus E, - J
\Bigr)
, (25)
which is unitary equivalent to \Delta n (22).
V. Let us turn our attention to the finding of the weight operators \{ mk\} n0 by the pair of functions
En(\lambda ), \widetilde En(\lambda ), for which Theorem 1 is true and En(0) = \widetilde En(0) = I. To do this, we by En(\lambda ),\widetilde En(\lambda ) construct the L. de Branges space \scrB (An, Bn) , in which we define the operator \widetilde An (23),
and then we construct the colligation \widetilde \Delta n and (25). Let Sn(w) = S\widetilde \Delta n
\bigl(
w - 1
\bigr)
, where S\widetilde \Delta n
(\lambda ) is the
characteristic function of \widetilde \Delta n (25). It is obvious that S1,1
n (w) = An(w)
\Bigl(
= 2 - 1
\Bigl(
En(w) + \widetilde En(w)
\Bigr) \Bigr)
and S1,2
n (w) = Bn(w)
\Bigl(
= (2i) - 1
\Bigl( \widetilde En(w) - En(w)
\Bigr) \Bigr)
, where
\Bigl\{
Sk,s
n (w)
\Bigr\} 2
1
are blocks Sn(w) cor-
responding to the decomposition E \oplus E. The remaining blocks Cn(w) = S2,1
n (w) and Dn(w) =
= S2,2
n (w) equal
Cn(w) =
w
2\pi
\int
\BbbR
I - A\ast
n(\lambda )
\lambda
(E\ast
n(\lambda ))
- 1E - 1
n (\lambda )
An(\lambda ) - An(w)
\lambda - w
d\lambda ,
Dn(w) = I +
w
2\pi
\int
\BbbR
I - A\ast
n(\lambda )
\lambda
(E\ast
n(\lambda ))
- 1E - 1
n (\lambda )
Bn(\lambda ) - Bn(w)
\lambda - w
d\lambda .
(26)
Theorem 4. Let there be determined two such matrix-functions En(\lambda ) and \widetilde En(\lambda ) that a) En(\lambda ),\widetilde En(\lambda ) are polynomials of degree n+ 1 with invertible leading coefficients; b) En(0) = \widetilde En(0) = I;
c) (12) take place; d) En(\lambda ) and \widetilde En(\lambda ) are invertible in \BbbC + and in \BbbC - correspondingly. Then by
the pair of functions An(\lambda ) =
1
2
( \widetilde En(\lambda ) + En(\lambda )), Bn(\lambda ) =
1
2i
( \widetilde En(\lambda ) - En(\lambda )) we can construct
the functions Cn(\lambda ), Dn(\lambda ) (26) such that the operator-function Sn(\lambda ) (5) has the J -properties.
Using the J -theory of V. P. Potapov [9, 10], we expand Sn(\lambda ) (5) into factors, then the simplest
factors ak(\lambda ) are the polynomials of the first degree and are given by (7), where mk \geq 0 and
mkJmk = 0, 0 \leq k \leq n.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
844 R. HATAMLEH, V. A. ZOLOTAREV
Lemma 2. Let m \geq 0 be an operator in E (\mathrm{d}\mathrm{i}\mathrm{m}E < \infty ) such that mJm = 0, where J is
given by (24), then
m =
\biggl[
PP \ast PQ\ast
QP \ast QQ\ast
\biggr]
,
besides, P, Q are linear bounded operators in E satisfying the condition P \ast Q = Q\ast P.
Proof. The lemma on block-matrix [7] yields that nonnegativity of the operator
m =
\biggl[
A B
B\ast C
\biggr]
\geq 0
is equivalent to the conditions: 1) A \geq 0; 2) solution X of the equation AX = B exists; 3) C -
- X\ast AX \geq 0, besides, the expression X\ast AX does not depend on X. Nonnegativity A \geq 0 signifies
that A = PP \ast (we can take P =
\surd
A as P ). Condition 2 yields that PP \ast X = B, therefore,
specifying Q\ast = P \ast X, we obtain B = PQ\ast .
The requirement mJm = 0 is equivalent to the equalities
AB\ast = BA, AC = B2, B\ast C = CB.
The first relation implies
0 = AB\ast - BA = PP \ast QP \ast - PQ\ast PP \ast = P \{ P \ast Q - Q\ast P\} P \ast ,
therefore \{ P \ast Q - QP \ast \} | P \ast E = 0 since the images P \ast and Q\ast belong to P \ast E. Taking into account
that i \{ P \ast Q - Q\ast P\} is self-adjoint and equals zero on its image, we obtain the desired condition
P \ast Q = Q\ast P.
To conclude the proof of lemma, it is left to show that C = QQ\ast . Condition 3 C - X\ast AX \geq 0
implies
0 \leq C - X\ast AX = C - X\ast B = C - X\ast PQ\ast = C - QQ\ast .
Existence of the solution of AX = B signifies that A is invertible on BE, therefore the equality
AC = B2 yields
C = XB = XPQ\ast = X\ast PQ\ast + (X - X\ast )PQ\ast = QQ\ast + (X - X\ast )PP \ast X,
and thus (X - X\ast )PP \ast X \geq 0. Since X\ast : PE \rightarrow QE and X : QE \rightarrow PE, the self-adjoint
operator (X - X\ast )PP \ast X maps the subspace \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ PE +QE\} onto itself. Note that 0 = P \ast Q -
- Q\ast P = P \ast (X - X\ast )P and thus the restriction of X - X\ast on PE equals zero. Therefore
to prove that (X - X\ast )PP \ast X = 0, it is necessary to ascertain that the image of the operator
(X - X\ast )PP \ast X belongs to PE. Let f \in \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ PE + QE\} and f \bot PE, then X : QE \rightarrow PE
implies
0 \leq \langle (X - X\ast )PP \ast Xf, f\rangle = - \langle X\ast PP \ast Xf, f\rangle = - \| P \ast Xf\| 2 ,
and thus P \ast Xf = 0 and (X - X\ast )PP \ast Xf = 0. Thus C = QQ\ast .
The condition B\ast C = CB holds automatically:
B\ast C - CB = QP \ast QQ\ast - QQ\ast PQ\ast = Q \{ P \ast Q - Q\ast P\} Q\ast = 0,
in virtue of P \ast Q = Q\ast P.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. II 845
Thus the operators mk in ak(\lambda ) (7) are always given by
mk =
\biggl[
PkP
\ast
k PkQ
\ast
k
QkP
\ast
k QkQk
\biggr]
, 1 \leq k \leq n. (27)
The “first line” Ln(\lambda ) = [An(\lambda ), Bn(\lambda )] of the operator-function Sn(\lambda ) (5) satisfies the recurrent
relation Ln(\lambda ) = Ln - 1(\lambda )an(\lambda ). Since a - 1
n (\lambda ) = I+i\lambda mnJ, then Ln - 1(\lambda ) = Ln(\lambda ) (I + i\lambda mnJ) ,
and thus
An - 1(\lambda ) = An(\lambda ) + \lambda Pn(\lambda )Q
\ast
n,
Bn - 1(\lambda ) = Bn(\lambda ) - \lambda Pn(\lambda )P
\ast
n ,
where Pn(\lambda ) is given by
Pn(\lambda )
df
= An(\lambda )Pn +Bn(\lambda )Qn. (28)
Repeating this step-by-step procedure, we find all the polynomials \{ Pk(\lambda )\} n0 , besides,
An(\lambda ) = I - \lambda
n\sum
k=0
Pk(\lambda )Q
\ast
k, Bn(\lambda ) = \lambda
n\sum
k=0
Pk(\lambda )P
\ast
k . (29)
Theorem 5. Let operator-functions En(\lambda ) and \widetilde En(\lambda ) satisfying the suppositions a) – d) of
Theorem 4 be given in a Hilbert space E. Then there exists such set of nonnegative operators
\{ mk\} n0 (27) in E \oplus E that P \ast
kQk = Q\ast
kPk, 0 \leq k \leq n, besides,
1) using the recurrent relation Lk - 1(\lambda ) = Lk(\lambda ) (I + i\lambda mkJ) we can construct the whole
Lk(\lambda ) = [Ak(\lambda ), Bk(\lambda )] , 0 \leq k \leq n, by the line Ln(\lambda ) = [An(\lambda ), Bn(\lambda )] (2An(\lambda ) = \widetilde En(\lambda ) +
+ En(\lambda ), 2iBn(\lambda ) = \widetilde En(\lambda ) - En(\lambda )), besides, \mathrm{d}\mathrm{e}\mathrm{g}Ak(\lambda ) = \mathrm{d}\mathrm{e}\mathrm{g}Bk(\lambda ) = k + 1;
2) for the functions Ek(\lambda ), \widetilde Ek(\lambda ) corresponding to Lk(\lambda ), the suppositions a) – d) of Theorem 4
are true;
3) the equalities (28) specify the system of polynomials \{ Pk(\lambda )\} n0 such that \mathrm{d}\mathrm{e}\mathrm{g}Pk(\lambda ) = k, the
leading coefficient of Pk(\lambda ) is invertible, Pk(0) = Pk, and the formulas (29) are true.
4. Jacobi operator. VI. Consider the “nth cut” of the Jacobi operator JE (3),
JE,n
df
=
\left[
C0 B\ast
0 0 \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot
B0 C1 B1 \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 \cdot \cdot \cdot Bn - 2 Cn - 1 B\ast
n - 1
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot 0 Bn - 1 Cn
\right] , (30)
which is self-adjoint in l2\BbbZ +(n)(E) (11). Using the form of operator V (10), we obtain
V JE,nf = xf(x) +Rn(x)fn, (31)
where f(x) = V f =
\sum n
k=1
Pk(x)fk \in \scrL n (10) (f = \mathrm{c}\mathrm{o}\mathrm{l} [f0, . . . , fn] \in l2\BbbZ +(n)(E)), and Rn(x) is a
polynomial of degree n+ 1,
Rn(x)
df
= Pn(x) [Cn - xI] + Pn - 1(x)B
\ast
n - 1, n \in \BbbZ +. (32)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
846 R. HATAMLEH, V. A. ZOLOTAREV
Taking into account (15), we write Rn(x) in the form Rn(x) = - Pn+1(x)Bn.
The operator JE,n (30), after the transform V, turns into the restriction of the operator of
multiplication by the independent variable in \scrL n,\Bigl( \widetilde JE,nf
\Bigr)
(x)
df
= P\scrL nxf(x) (V JE,n = \widetilde JE,nV ),
where P\scrL n is the orthoprojector on \scrL n.
Calculate the resolvent (JE,n - wI) - 1 , and let g(w) = (JE,n - wI) - 1 f, then JE,ng(w) -
- wg(w) = f. Using (31), we obtain
(x - w)g(x,w) +Rn(x)gn(w) = f(x),
where f(x) = V f =
\sum n
k=0
Pk(x)fk and g(x,w) = V g(w) =
\sum n
k=0
Pk(x)gk(w). Supposing in
this equality that x = w, we find Rn(w)gn(w) = f(w). If w \in \BbbC is such that Rn(w) is invertible
(which is possible in view of the invertibility of the leading coefficient of Rn(x)), then from the last
equality we obtain
g(x,w) =
\Bigl( \widetilde JE,n - wI
\Bigr) - 1
f(x) =
f(x) - Rn(x)R
- 1
n (w)f(w)
x - w
. (33)
Therefore the “kth” component gk(w) of the vector g(w) = (JE,n - wI) - 1 f equals
gk(w) =
\int
\BbbR
P \ast
k (x)dF (x)
\biggl\{
f(x) - Rn(x)R
- 1
n (w)f(w)
x - w
\biggr\}
, 0 \leq k \leq n. (34)
Theorem 6. The resolvent of the operator \widetilde JE,n (33) is given by (34), and the resolvent of the
operator JE,n (30) is given by formula (47), where f(x) = V f (10), gk(w) is the “kth” component
of the vector g(w), besides, Rn(w) equals (32).
We can write the L. de Branges map \scrB (18) from l2\BbbZ +(n)(E \oplus E,m) in \scrB (An, Bn) in the form
F (\lambda ) = \scrB f =
n\sum
k=0
Pk(\lambda ) [P
\ast
k (0)uk +Q\ast
k(0)vk]
since Ak - 1(\lambda )Pk(0) + Bk - 1(\lambda )Qk(0) = Pk(\lambda ), 1 \leq k \leq n, where f = [f0, . . . , fn] \in l2\BbbZ +(n)(E \oplus
\oplus E,m) (fk = \mathrm{c}\mathrm{o}\mathrm{l} [uk, vk] , 0 \leq k \leq n). After the transform \scrB (18), the Jacobi operator JE,n (30)
equals \Bigl( \widehat JE,nF
\Bigr)
(\lambda ) = P\scrB (An,Bn)\lambda F (\lambda ),
where F (\lambda ) \in \scrB (An, Bn) and P\scrB (An,Bn) is the orthoprojector on \scrB (An, Bn) . Similarly to (34), the
formula \Bigl( \widehat JE,n - wI
\Bigr) - 1
F (\lambda ) =
F (\lambda ) - Rn(\lambda )R
- 1
n (w)F (w)
\lambda - w
is true, besides, Rn(\lambda ) is given by (32).
Let \mathrm{d}\mathrm{i}\mathrm{m}E = r < \infty . Let us find the eigenvectors JE,nf = \lambda f of the operator JE,n (30), where
f = \mathrm{c}\mathrm{o}\mathrm{l} [f0, . . . , fn] \in l2\BbbZ +(n)(E) (11). Then
f0 = P \ast
0 (\lambda )h, . . . , fn = P \ast
n(\lambda )h, R\ast
n(\lambda )h = 0,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. II 847
where the function Rn(\lambda ) is given by (32). The last of the equations R\ast
n(\lambda )h = 0 defines the
eigenvalues \{ \lambda k\}
r(n+1)
1 , \lambda k \in \BbbR , of the operator JE,n which are the zeros of the polynomial
\mathrm{d}\mathrm{e}\mathrm{t}R\ast
n(\lambda ) = 0 and hk \in \mathrm{K}\mathrm{e}\mathrm{r}R\ast
n (\lambda k) . Therefore f\lambda k
= \mathrm{c}\mathrm{o}\mathrm{l} [P \ast
0 (\lambda k)hk, . . . , P
\ast
n (\lambda k)hk] are the
eigenvectors of JE,n, besides, f\lambda k
\bot f\lambda s as \lambda k \not = \lambda s. When \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{K}\mathrm{e}\mathrm{r}R\ast
n (\lambda k) = lk > 1, it is easy to
construct such a basis \{ hsk\}
lk
1 in \mathrm{K}\mathrm{e}\mathrm{r}R\ast
n (\lambda k) that the vectors fs
\lambda k
= \mathrm{c}\mathrm{o}\mathrm{l} [P0 (\lambda k)h
s
k, . . . , Pn (\lambda k)h
s
k] ,
1 \leq s \leq lk, are orthogonal. To do this one should use the Hilbert – Schmidt orthogonalization. Let
the vectors hk and \^hk from \mathrm{K}\mathrm{e}\mathrm{r}R\ast
n (\lambda k) correspond to f\lambda k
and \^f\lambda k
from \mathrm{K}\mathrm{e}\mathrm{r} (JE,n - \lambda kI) , then
h\prime k = \^hk - \mu khk corresponds to the linear span f \prime
\lambda k
= \^f\lambda k
- \mu kf\lambda k
, \mu k \in \BbbC . Then from f \prime
\lambda k
\bot f\lambda k
we find \mu k =
\Bigl\langle
Kn (\lambda k, \lambda k) \^hk, hk
\Bigr\rangle
\{ \langle Kn (\lambda k, \lambda k)hk, hk\rangle \} - 1 . Using this technique the required
number of times, we obtain an orthogonal set of vectors in \{ fs
\lambda k
\} l1 in \mathrm{K}\mathrm{e}\mathrm{r} \{ JE,n - \lambda kI\} . So, the
orthogonal basis \{ fs
\lambda k
\} of the eigenvectors of JE,n in l2\BbbZ +(n)(E) (11) exists.
Theorem 7. Let \mathrm{d}\mathrm{i}\mathrm{m}E = r < \infty and \{ \lambda k\} be zeros of the polynomial \mathrm{d}\mathrm{e}\mathrm{t}R\ast
n(\lambda ) = 0, where
Rn(\lambda ) is given by (32), besides, \mathrm{K}\mathrm{e}\mathrm{r}R\ast
n (\lambda k) = Lk and \mathrm{d}\mathrm{i}\mathrm{m}Lk = lk > 1. Suppose that \{ hsk\}
lk
1
from Lk are such that
\bigl\langle
Kn (\lambda k, \lambda k)h
s
k, h
p
k
\bigr\rangle
= 0 as s \not = p. Then the reproducing kernel Kn(\lambda ,w)
(9) generates the orthogonal basis
\{ Kn (x, \lambda k)h
s
k\} , 1 \leq s \leq lk, 1 \leq k \leq (n+ 1)r),
in the space \scrL n (8), besides, every function (34) is an eigenfunction for \widetilde JE,n (33) and corresponds
to the eigenvalue \lambda k.
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 selfadjoint 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.
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P. 245 – 267.
8. Lopez-Rodriguez P. Riesz’s theorem for orthogonal matrix polynomials // Const. Approxim. – 1999. – 15, № 1. –
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Received 06.10.15,
after revision — 10.03.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
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| id | umjimathkievua-article-1738 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:43Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7b/63eb13941181f6440c433596ea5d147b.pdf |
| spelling | umjimathkievua-article-17382019-12-05T09:25:15Z Jacobi operators and orthonormal matrix-valued polynomials. II Оператори Якобi та ортогональнi операторнозначнi полiноми. ІІ Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. We use a system of operator-valued orthogonal polynomials to construct analogs of L. de Branges spaces and establish their relationship with the theory of nonself-adjoint operators. Iз використанням системи операторнозначних ортогональних полiномiв побудовано аналоги просторiв Л. де Бранжа та встановлено їх зв’язок з теорiєю несамоспряжених операторiв. Institute of Mathematics, NAS of Ukraine 2017-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1738 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 6 (2017); 836-847 Український математичний журнал; Том 69 № 6 (2017); 836-847 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1738/720 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. II |
| title | Jacobi operators and orthonormal matrix-valued polynomials. II |
| title_alt | Оператори Якобi та ортогональнi
операторнозначнi полiноми. ІІ |
| title_full | Jacobi operators and orthonormal matrix-valued polynomials. II |
| title_fullStr | Jacobi operators and orthonormal matrix-valued polynomials. II |
| title_full_unstemmed | Jacobi operators and orthonormal matrix-valued polynomials. II |
| title_short | Jacobi operators and orthonormal matrix-valued polynomials. II |
| title_sort | jacobi operators and orthonormal matrix-valued polynomials. ii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1738 |
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