DOI 10.1007/s11253-017-1360-4
Ukrainian Mathematical Journal, Vol. 69, No. 2, July, 2017 (Ukrainian Original Vol. 69, No. 2, February, 2017)
JACOBI OPERATORS AND ORTHONORMAL MATRIX-VALUED POLYNOMIALS. I
R. Hatamleh1 and V. A. Zolotarev2
UDC 517.9
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.
Introduction
Jacobi matrix is the canonical form of a self-adjoint operator with simple spectrum [1]. The spectral analysis
of this matrix is closely connected with the study of orthogonal polynomials [2, 3]. This field of analysis has deep
connections with moment problem, interpolation problems, issues of extension of symmetric operators, etc. [2, 3].
The present work develops the investigations 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 a block Jacobi operator (is unitarily
equivalent to it). Moreover, the 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) were studied in the works [4–8]. Important results in the spectral analysis obtained
in [4, 5] found their extensive application in the problem of moments. The generalization of the scalar case [2] to
the matrix-valued case was studied in [7, 8] and represented in the comprehensive survey [6]. The establishment
of links between the block Jacobi matrices and the theory of nonself-adjoint operators with analytic analogs of the
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 by using an introduced notion of nondegenerateness of measure (see an analog
in [6]) and then an operator-valued function with J-properties is constructed and its multiplicative expansion is
obtained by the methods of J-theory of Potapov [9, 10].
The constructions used in the present paper refer to the so-called ”truncated” problem (n 2 N), i.e., to the
finite block Jacobi matrix.
1. Block Jacobi Operators
I. Consider the spectral resolution [1]
A=
Z
λdEλ
(1)
R
of a linear self-adjoint operator A given in a separable Hilbert space H.
We now select a vector f1 from a set dense in H (in view of the property of separation) and construct
a subspace
df
L(f1 ) = span{E∆ f1 : ∆ 2 R},
(2)
1
Jadara University, Irbid, Jordan.
Institute for Low Temperature Physics and Engineering , Ukrainian National Academy of Sciences and Karazin Kharkov National University, Kharkiv, Ukraine.
2
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 2, pp. 228–239, February, 2017. Original article submitted October 2, 2015,
revision submitted March 10, 2016.
0041-5995/17/6902–0269
c 2017
Springer Science+Business Media New York
269
R. H ATAMLEH AND V. A. Z OLOTAREV
270
where ∆ runs over the totality of all intervals of the axis R. Clearly, L1 (f ) and its orthogonal complement
H1 = H
L (f1 )
are Et -invariant (t 2 R). Since H1 is also separable, selecting f2 from a countable dense set in H1 , we define a subspace L (f2 ) given by (2) in H1 . Repeating this procedure of removal L (fk ) countably many times,
we obtain
H=
1
X
⊕L(fk ).
(3)
k=1
Note that this procedure can be terminated after finitely many steps and even in the first step. The latter leads
to the representation by the classical Jacobi matrix with scalar entries. We define the generating subspace [1] G
for the operator A as follows:
df
G = span{fk : k 2 N}.
(4)
H = span{E∆ g : g 2 G; ∆ 2 R},
(5)
Thus, relations (3) implies that
where, as usual, ∆ belongs to the set of all intervals from R. Consider a Hilbert space E (dim E ≥ dim G) and
assume that ψ is a linear bounded operator from E onto G. We define a nondecreasing operator function in E
df
F (x) = ψ ⇤ Ex ψ,
x 2 R.
(6)
As ψ, we can take,�e.g., the orthoprojector
PG on G and suppose that E = G.
�
Denote by L2R E, dF (x) the Hilbert space of E-valued vector functions on R,
9
8
Z
=
⌦
↵
� df <
LR E, dF (x) = f (x) :
dF (x)f (x), f (x) E < 1
;
:
�
2
(7)
R
generated by the closure of the linear span of finite continuous functions f (x) with subsequent factorization by the
kernel of metrics (7). This definition is correct (see [3, 11]). We specify a linear operator U :
�
U : L2R E, dF (x)
�
! H,
f = U f (x),
df
f =
Z
dEx ψf (x).
R
The image of the operator U is dense in H because the vectors
Z
dEx ψχ∆ (x) = E∆ ψf,
f 2 E,
∆ 2 R,
R
whose linear span is dense in H belong to it in view of (5) (ψf = g 2 G).
(8)
JACOBI O PERATORS AND O RTHONORMAL M ATRIX -VALUED P OLYNOMIALS . I
271
�
�
If g(x) 2 L2R E, dF (x) and f is given by (8), then
⌦
↵
Et f, ψg(x) =
Zt
−1
⌦
↵
dF (s)f (s), g(x) .
(9)
�
�
Assume that f (x) from (8) is differentiable and f 0 (x) 2 L2R E, dF (x) . Hence, in view of (9),
2
kf k =
Z
⌦
↵
f, dEx ψf (x)
R
=
Z
⌦
↵
d f, Ex ψf (x) −
R
=
Z
d
R
=
Zx
Z
R
⌦
↵
Ex f, ψf 0 (x) dx
hdF (t)f (t), f (x)i −
R −1
−1
Z
⌦
R
Z Zx
⌦
↵
dF (t)f (t), f 0 (x) dx
↵
dF (x)f (x), f (x) .
�
�
Therefore, U (8) is isometric on a dense set in L2R E, dF (x) 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 L2R E, dF (x) . Then,
for all h 2 H,
Z
⌦
↵
hAf, hi = x f, dEx h
R
=
Z
R
=
Z
R
x
Z
R
xd
⌦
↵
dEs ψf (s), dEx h =
Zx
−1
⌦
↵
dEs ψf (s), h =
Z
E
Z
Z
xdx
⌦
↵
dEx ψxf (x), h .
R
R
⌦
↵
dEs ψf (s), Ex h
Consequently,
Af =
Z
dEx ψ(xf (x)),
(10)
R
�
�
and, hence, AU = U Q, where Q is the operator of multiplication by the independent variable in L2R E, dF (x) ,
i.e.,
df
(Qf )(x) = xf (x)
�
�
(f (x) 2 L2R E, dF (x) .
(11)
R. H ATAMLEH AND V. A. Z OLOTAREV
272
The isometry of U (8) and (10) imply that
kAf k2 =
Z
R
⌦
↵
x2 dF (x)f (x), f (x) = kQf (x)k2 .
Therefore, f (x) belongs to the domain DQ of the operator Q (11) if and only if f (8) belongs to the domain DA
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 L2R E, dF (x) (7), AU = U Q, where U is given by (8); Ex is the resolution of identity for
the operator A; F (x) is given by formula (6); and ψ is a linear bounded operator from E on the generating
subspace G (4).
II. Let A be a bounded self-adjoint operator. Then
n
A ψg =
Z
xn dEx ψg
R
is meaningful for all g 2 G and all n 2 Z+ . We now show that the linear span of these vectors is dense in H.
If a vector f 2 H is such that f ? An ψg (for all g 2 G and all n 2 Z+ ), then, by using representation (8) for f,
we obtain
Z
⌦
↵
n
dF (x)xn g, f (x) .
0 = hA ψg, f i =
R
Therefore f (x) ? Pn (x), where P�n (x) is an �arbitrary E-valued polynomial of degree n. Since the set of polynomials of this kind is dense in L2R E, dF (x) (dF (x) has the dense support), we conclude that f (x) = 0 and,
therefore, f = 0.
Theorem 2. For every bounded self-adjoint operator A acting in a separable Hilbert space H,
�
H = span An ψg : g 2 E, n 2 Z+ .
(12)
In the case of unboundedness of A, see [1, 2].
We define subspaces
�
df
Hn = span Ak ψg : g 2 E; 0 k n ,
n 2 Z+ ,
(13)
which are ordered by inclusion, Hk ✓ Hs as s > k. Assume that
df
G n = Hn
Hn−1 ,
n 2 Z+ ,
(14)
where H−1 = {0} and G0 = G (4). Then
H=
1
X
k=0
⊕Gk .
(15)
JACOBI O PERATORS AND O RTHONORMAL M ATRIX -VALUED P OLYNOMIALS . I
273
If gk 2 Gk , k 2 Z+ , then hAgk , gs i = 0 as s > k + 1 since Agk 2 Hk+1 . Similarly, hAgk , gs i = hgk , Ags i
as k > s + 1 (Ags 2 Hs+1 ). Hence,
hAgk , gs i = 0 as
s>k+1
s < k − 1,
and
k, s 2 Z+ .
Therefore, the operator A has a three-diagonal block structure corresponding to expansion (15):
2
e0
A
6
6B
⇤
6 e0
6
6
A=6 0
6
6
6· · ·
4
···
e0
B
0
0
e1
A
e1
B
0
···
···
···
···
···
···
e⇤
B
1
e2
A
e2
B
···
3
7
· · ·7
7
7
7
· · ·7,
7
7
· · ·7
5
(16)
···
where
ek = Pk APk : Gk ! Gk
A
and
ek = Pk APk+1 : Gk+1 ! Gk
B
[Pk is the orthoprojector on Gk (14), k 2 Z+ ]. The definition of Gk (14) implies that dim Gk dim G, k 2 Z+ .
For dim Gk = dim G we specify unitary operators Vk : Gk ! G. If dim Gk < dim G, then we can define
ek V ⇤ , Bk = Vk B
ek V ⇤ ,
isometric operators Vk : Gk ! G. We now consider a set of operators in G : Ak = Vk A
k
k+1
k 2 Z+ , and define the block Jacobi operator
2
3
0 ···
A0 B0 0
6
7
6B⇤ A B
7
0
·
·
·
1
1
6 0
7
6
7
df 6
7
⇤
J G = 6 0 B 1 A2 B 2 · · · 7 ,
(17)
6
7
6
7
6· · · · · · · · · · · · · · ·7
4
5
··· ··· ··· ··· ···
in the Hilbert space lZ2 + (G). Let V = diag[I, V1 , . . .] be an isometric (by construction) operator from H (15) into
lZ2 + (G). Then V A = JG V.
Theorem 3. Any bounded self-adjoint operator acting in a separable Hilbert space H is isometrically equivalent to the Jacobi operator JG (17) in the space lZ2 + (G).
2. Matrix-Valued Orthogonal Polynomials
III. The spectral analysis of Jacobi matrices (dim G = 1) is closely connected with the properties of orthogonal polynomials [1–6]. We now proceed to the construction of matrix-valued orthogonal polynomials.
Definition 1. We say that a measure dF (x) satisfies the nd-condition (non degenerata) if, for every E-valued
polynomial of finite degree
Pn (x) =
n
X
k=0
xk g k ,
gk 2 E,
1 k n,
2 Z+ ,
R. H ATAMLEH AND V. A. Z OLOTAREV
274
the estimate
Z
R
⌦
n
X
↵
kgk k2 ,
dF (x)Pn (x), Pn (x) > δn
(18)
k=0
holds and, in addition, the number δn does not depend on the vectors {gk }n1 , and δn > 0 for all n 2 Z+ .
Note that the nd-condition is per se equivalent to the nondegenerateness of a 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 dim E = r < 1, then the vector
An ψg does not belong to the space Hn−1 (13) for any g 2 E and n 2 N.
Proof. Assume the contrary, i.e., suppose that there exists a vector g 2 E such that An ψg 2 Hn−1 for some
n 2 N. Then
n
A ψg +
n−1
X
Ak ψgk = 0,
gk 2 E,
0 k n − 1.
k=1
This means that
Z
dEt ψPn (t) = 0,
R
where
Pn (t) = tn g +
n−1
X
tk g k
k=0
is an E-valued polynomial. Applying Ex to this equality, we obtain
Zx
dEt ψPn (t) = 0 8x 2 R
−1
and, therefore,
Zx
−1
⌦
↵
dEt ψPn (t), ψf =
Zx
−1
⌦
↵
dF (t)Pn (t), f = 0 8f 2 E.
Consequently,
0=
Z
R
ϕ(x)d
Zx
0
⌦
↵
dF (t)Pn (t), f =
Z
R
⌦
↵
dF (x)Pn (x), ϕ(x)f ,
for any scalar function ϕ(x): R ! C. Hence,
Z
R
⌦
↵
dF (x)Pn (x), f (x) = 0,
JACOBI O PERATORS AND O RTHONORMAL M ATRIX -VALUED P OLYNOMIALS . I
275
where f (x) is any function of the form
f (x) =
m
X
ϕk (x)fk ,
m 2 Z+ .
k=0
Assume that f (x) = Pn (x). This yields
Z
R
⌦
↵
dF (x)Pn (x), Pn (x) = 0,
which contradicts the nd-condition.
Theorem 4 is proved.
This theorem implies that the vector
f = An ψg − PHn−1 An ψg
⇥
⇤
is nonzero for all g 2 E and all n 2 N PHn−1 is the orthoprojector on Hn−1 (13) . It is obvious that PHn−1 f = 0.
Consequently, f ? Hs (8s, 0 s n − 1). Since
f = An ψg +
n−1
X
Ak ψgk ,
(19)
k=0
each g 2 E (dim E = r < 1) is associated with the set of vectors {gk }n−1
from E. Relation (19) follows
0
df
from (13) for dim E < 1. This correspondence defines the linear operators Nk g = gk , 0 k n − 1. Since
dim E < 1, Nk are bounded for all k.
We write the vector f (19) in the form
Z
f = dEt ψ Pen (t)g,
R
where
Pen (t) = tn + tn−1 Nn−1 + . . . + N0 .
The orthogonality f ? Hs , 0 s n − 1 means that
Z
R
Pes⇤ (t)dF (t)Pen (t) = 0,
0 s n − 1.
The operator
Dn =
Z
R
Pen (t)dF (t)Pen (t)
is nonnegative and invertible because the nd-condition (18) implies that
� 1 �2
� 2 �
�Dn g � > δn kgk2 .
R. H ATAMLEH AND V. A. Z OLOTAREV
276
−
1
Therefore the polynomial Pn (t) = Pen (t)Dn 2 is normalized to unity:
Z
Pn⇤ (t)dF (t)Pn (t) = IE .
R
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), Ex is the resolution of identity for the operator A, and dim E = r < 1), then there exists
a family of matrix-valued (in E) polynomials {Pn (x)}1
0 , such that
Z
Pk⇤ (x)dF (x)Pn (x) = δk,n IE ,
k, n 2 Z+ .
(20)
R
Moreover, deg Pn (x) = n and the leading coefficient of Pn (x) is invertible (8n 2 Z+ ).
The expansion [2–6]
xPn (x) = Pn+1 (x)Bn(n+1) + Pn (x)Bn(n) + . . . + P0 (x)Bn(0) ,
(s)
(s)
where Bn are linear bounded operators in E, 0 s n + 1, and (20) imply that Bn = 0 for 0 s n − 2.
Moreover,
Bn(n+1) =
Z
⇤
xPn+1
(x)dF (x)Pn (x),
Bn(n) =
R
Z
xPn⇤ (x)dF (x)Pn (x),
R
Bn(n−1) =
Z
⇤
(x)dF (x)Pn (x);
xPn−1
R
� (n) �⇤
� (n) �⇤
�
1
(n−1)
(n)
and, thus, Bn = Bn
and Bn
= Bn−1 . Hence, the totality Pn (x) 1 is the solution of the finitedifference equation
⇤
,
xPn (x) = Pn+1 (x)Bn + Pn (x)Cn + Pn−1 (x)Bn−1
df
(n+1)
n 2 Z+ ,
(21)
(n)
where P−1 (x) = 0, Bn = Bn
, and Cn = Bn , n 2 Z+ . The invertibility of the leading coefficients of
the polynomials Pn (x) implies the invertibility of all operators Bn . Therefore, the polynomials of the first kind
Pn (x) are unambiguously found as the solutions of (21) if we take into account the initial conditions
P0 (x) = D0 ,
P1 (x) = D0 (xI − C0 )B0−1 ,
(22)
�
�−1/2
is an invertible positive operator. The expression
where D0 = F (1) − F (−1)
df
Qn (x) =
Z
R
dF (ξ)
Pn (ξ) − Pn (x)
,
ξ−x
n 2 Z+ ,
(23)
JACOBI O PERATORS AND O RTHONORMAL M ATRIX -VALUED P OLYNOMIALS . I
277
defines [2] operator-valued polynomials of the second kind deg Qn (x) = n − 1. Furthermore, Qn (x) also satisfy
the finite-difference equation (21) and the initial data
Q1 (x) = D0−1 B0−1 .
Q0 (x) = 0,
(24)
We now construct a Jacobi operator
2
C0
6
6B
6 0
6
df 6
JE = 6 0
6
6
6· · ·
4
···
B0⇤
0
0
C1
B1⇤
0
B1
C2
B2⇤
···
···
···
···
···
···
···
3
7
· · ·7
7
7
7
· · ·7
7
7
· · ·7
5
···
(25)
by the coefficients {Bn , Cn }1
0 from (21). Then the recurrence relations (21) formally implies that
P(x)JE = xP(x),
Q(x)JE = xQ(x),
(26)
where
⇥
⇤
P(x) = P0 (x), P1 (x), . . .
⇥
⇤
and Q(x) = Q0 (x), Q1 (x), . . . .
Let Yn = Yn (λ) and Zn = Zn (w) be the solutions of (21) corresponding to λ and w, respectively (λ,
w 2 C). The Green formula [2–4, 6]
(λ − w̄)
n
X
⇤
⇤
⇤
⇤
− Ym Bm−1 Zm−1
+ Ym−1 Bm−1
Zm
Yk Zk⇤ = Yn+1 Bn Zn⇤ − Yn Bn⇤ Zn+1
(27)
k=m
is true for all n, m 2 N. In particular, for m = 1, if Yk = Pk (λ) and Zk = Pk (w), then, in view of (22), we
obtain the Christoffel–Darboux formula [2–4, 6]
(λ − w̄)
n
X
⇤
(w)
Pk (λ)Pk⇤ (w) = Pn+1 (λ)Bn Pn⇤ (w) − Pn (λ)Bn⇤ Pn+1
(28)
k=0
for all n 2 Z+ . Assuming in (27) that m = 1, Yk = Pk (λ), and Zk = Qk (w) and using (22) and (24), we arrive
at the equality
(λ − w̄)
n
X
Pk (λ)Q⇤k (w) = Pn+1 (λ)Bn Q⇤n (w) − Pn (λ)Bn⇤ Q⇤n+1 (w) + IE
8n 2 Z+ .
(29)
k=0
Finally, the relation
(λ − w̄)
n
X
Qk (λ)Q⇤k (w) = Qn+1 (λ)Bn Q⇤n (w) − Qn (λ)Bn⇤ Q⇤n+1 (w)
k=0
follows from (27) with m = 1, Yk = Qk (λ), and Zk = Qk (w).
(30)
R. H ATAMLEH AND V. A. Z OLOTAREV
278
Lemma 1. If Pn (λ) and Qn (λ) are the solutions of the finite-difference equation (21) satisfying the conditions (22) and (24), then the Liouville–Ostrogradskii formula [2–4, 6]
is true and, in addition,
�
Pn⇤ (λ̄)Qn+1 (λ) − Q⇤n (λ̄)Pn+1 (λ) Bn = IE
Pn⇤ (λ̄)Qn (λ) − Q⇤n (λ̄)Pn (λ) = 0
8n 2 Z+
8n 2 Z+ .
(31)
(32)
Proof. We prove both equalities (31) and (32) simultaneously by induction on n. For n = 0, the validity
of (31), (32) follows from the initial data (22) and (24). Assume that the statement is proved for all k (0 k n).
It is necessary to show that this yields (31) and (32) for k = n + 1. By using (21) and the invertibility of Bn ,
we obtain (by the induction hypothesis):
⇥ ⇤
⇤
Bn⇤ Pn+1
(λ̄)Qn+1 (λ) − Q⇤n+1 (λ̄)Pn+1 (λ) Bn
⇥
⇤⇥
⇤
⇤
⇤
= (λ − Cn )Pn⇤ (λ̄) − Bn−1 Pn−1
(λ̄) Qn (λ)(λ − Cn ) − Qn−1 (λ)Bn−1
⇤
⇤⇥
⇥
⇤
− (λ − Cn )Q⇤n (λ̄) − Bn−1 Q⇤n−1 (λ̄) Pn (λ)(λ − Cn ) − Pn−1 (λ)Bn−1
⇥
⇤ ⇤
= (λ − Cn ) Pn⇤ (λ̄)Qn−1 (λ) − Q⇤n (λ̄)Pn−1 (λ) Bn−1
Similarly,
⇥
Q.E.D.
⇥ ⇤
⇤
− Bn−1 Pn−1
(λ̄)Qn (λ) − Q⇤n−1 (λ̄)Pn (λ) (λ − Cn ) = 0.
⇤
⇤
Pn+1
(λ̄)Qn+2 (λ) − Q⇤n+1 (λ̄)Pn+2 (λ) Bn+1
⇥
⇤
⇥
⇤
⇤
= Pn+1
(λ̄) Qn+1 (λ)(λ − Cn+1 ) − Qn (λ)Bn − Q⇤n+1 (λ̄) Pn+1 (λ)(λ − Cn+1 ) − Pn (λ)Bn
�
⇤
(λ̄)Qn (λ) Bn⇤ = I,
= Q⇤n+1 (λ̄)Pn (λ) − Pn+1
IV. By using the polynomials Pn (λ) and Qn (λ), we construct an operator function
df
Wn (λ) =
"
Pn (λ)
Pn+1 (λ)Bn
Qn (λ) Qn+1 (λ)Bn
#
.
(33)
Moreover, deg Wn (λ) = n and we define the involution J in E ⊕ E as follows:
df
J =
"
0
iIE
−iIE
0
#
.
(34)
JACOBI O PERATORS AND O RTHONORMAL M ATRIX -VALUED P OLYNOMIALS . I
279
It follows from formulas (28)–(30) that
"
#"
#
n
Pk (λ) 0 Pk⇤ (w) Q⇤k (w)
X
λ
−
w̄
Wn (λ)JWn⇤ (w) − J =
i
0
0
n=0 Qk (λ) 0
(35)
and, hence, Wn (λ) (33) has the J-properties [6, 7]:
8
>
≥ 0,
>
>
>
<
Wn (λ)JWn⇤ (λ) − J = = 0,
>
>
>
>
: 0,
Equation (21) implies that
λ 2 C+ ,
λ 2 R,
(36)
λ 2 C− .
Wn (λ) = Wn−1 (λ)bn (λ),
(37)
where
2
bn (λ) = 4
0
⇤−1
Bn−1
−1
Bn−1
Bn−1 (λI − Cn )
3
5,
n 2 Z+ ,
(38)
df
in view of the fact that B−1 = I. Therefore
y
Wn (λ) = U0
n
Y
bk (λ),
(39)
k=1
where U0 is J-unitary,
df
2
U0 = 4
0
P0 (λ)
−P0−1 (λ)
0
3
5,
and, hence, we can assume that W−1 (λ) = U0 . The “primary” factors bk (λ) (38) also have the J-properties (36)
because
λ − w̄
bk (λ)Jb⇤k (w) − J =
i
"
#
0
� −1 �⇤ ,
−1
Bk−1
0 Bk−1
0
λ − w̄
b⇤k (w)Jbk (λ) − J =
i
"
0
0
0 IE
#
for all k 2 Z+ . Thus, factorization (39) is realized in the context of the class of operator functions satisfying rela-
R. H ATAMLEH AND V. A. Z OLOTAREV
280
tions (36). Moreover, the factors bk (λ) (38) are constructed according to the elements of the Jacobi matrix JE (25).
A factorization similar to (38) was obtained in [9] in a somewhat different form.
Relation (35) implies that
� �
Wn (λ)JWn⇤ λ̄ J = IE⊕E ,
Therefore, the operator
� �3
� �
⇤P ⇤
⇤ Q⇤
B
λ̄
−B
n n+1 λ̄
n n+1
� �
df
5,
Wn−1 (λ) = JWn⇤ λ̄ J = 4
� �
� �
⇤
−Qn λ̄
Pn λ̄
2
n 2 Z+ ,
(40)
is the right inverse for Wn (λ) (33). The fact that Wn−1 (λ) (40) is also the left inverse for Wn (λ) follows from
(31) and (32).
Observation 1. Relations (28)–(30) provide Wn (λ) (33) with the J-properties (36) and also secure the
existence of the right inverse Wn−1 (λ) (40); the equalities (31), (32) are equivalent to Wn−1 (λ)Wn (λ) = I.
Hence, the relations (28)–(30) and (31), (32) for the polynomials Pn (λ) and Qn (λ) admit a natural interpretation
in terms of the J-properties of the function Wn (λ).
The operator function
df
Sn (λ) = Wn (λ)Wn−1 (0)
(41)
2
3
32
⇤ (w) Q⇤ (w)
n
P
(λ)
0
P
k
X
k
k
λ − w̄
4
5,
54
Sn (λ)JSn⇤ (w) − J =
i
0
0
k=0 Qk (λ) 0
(42)
also has the J-properties (36) and
by virtue of the J-unitarity of Wn−1 (0). The function Sn (λ) is such that Sn (0) = I. It is easy to see that Sn (λ)
is equal to
2
3
An (λ) Bn (λ)
5,
Sn (λ) = 4
(43)
Cn (λ) Dn (λ)
where
df
An (λ) = Pn (λ)Bn⇤ Q⇤n+1 (0) − Pn+1 (λ)Bn Q⇤n (0) = I − λ
n
X
Pk (λ)Q⇤k (0),
k=0
df
⇤
Bn (λ) = Pn+1 (λ)Bn Pn⇤ (0) − Pn (λ)Bn⇤ Pn+1
(0) = λ
n
X
Pk (λ)Pk⇤ (0),
k=0
df
Cn (λ) = Qn (λ)Bn⇤ Q⇤n+1 (0) − Qn+1 (λ)Bn Q⇤n (0) = −λ
(44)
n
X
Qk (λ)Q⇤k (0),
k=0
df
⇤
Dn (λ) = Qn+1 Bn Pn⇤ (0) − Qn (λ)Bn⇤ Pn+1
(0) = 1 + λ
n
X
k=0
Qk (λ)Pk⇤ (0),
JACOBI O PERATORS AND O RTHONORMAL M ATRIX -VALUED P OLYNOMIALS . I
281
in view of the form of Wn (λ) (33) and Wn−1 (0) (40) and also of (28)–(32). Functions (44) are similar to the wellknown scalar [2, 3] and matrix [7] functions and, in addition, deg An (λ) = deg Bn (λ) = n + 1 and deg Cn (λ) =
deg Dn (λ) = n.
Observation 2. Functions (44), despite the properties following from (42), satisfy the equalities
A⇤n (λ̄)Dn (λ) − Cn⇤ (λ̄)Bn (λ) = IE ,
Dn⇤ (λ̄)Bn (λ) = Bn⇤ (λ̄)Dn (λ),
(45)
A⇤n (λ̄)Cn (λ) = Cn⇤ (λ̄)An (λ),
n 2 Z+ ,
which follow from the fact that JSn⇤ (λ̄)J is the left inverse for Sn (λ). Relations (45) can be proved directly by
using (31) and (32).
df
Observation 3. The normalization Et at zero for Sn (λ) (41) is not binding. If we consider Sn (λ, λ0 ) =
Wn (λ)Wn−1 (λ0 ), λ0 2 R, then:
(i) Sn (λ0 , λ0 ) = IE⊕E ;
(ii) relation (42) takes place by virtue of the J-unitarity of Wn−1 (λ0 );
(iii) for Sn (λ, λ0 ), representation (43) is true with a proper version of relations (44).
Relations (37) and (41) imply that
Sn (λ) = Sn−1 (λ)an (λ),
n 2 Z+ ,
(46)
an (λ) = I − iλmn J,
n 2 Z+ .
(47)
Pn (0)Pn⇤ (0)
3
where the factor an (λ) = Wn−1 (0)bn (λ)Wn−1 (0) is equal to
df
Moreover,
df
2
mn = 4
Pn (0)Q⇤n (0)
Qn (0)Pn⇤ (0) Qn (0)Q⇤n (0)
5 ≥ 0,
n 2 Z+ ,
(48)
and mn Jmn = 0 in view of (32). Relation (46) implies that
y
Sn (λ) =
n
Y
ak (λ).
(49)
k=0
In addition, S−1 (λ) = I (W−1 (λ) = U0 ). The factors ak (λ) (47) have the J-properties because
ak (λ)Ja⇤k (w) − J =
λ − w̄
mk ,
i
a⇤k (w)Jak (λ) − J =
λ − w̄
Jmk J.
i
The formula mk Jmk = 0 implies that ak (λ) (47) have the exponential representation
ak (λ) = exp{−iλmk J},
k 2 Z+ .
(50)
R. H ATAMLEH AND V. A. Z OLOTAREV
282
Theorem 6. The operator function Wn (λ) (33) has the J-properties (36) and admits the multiplicative expansion (39), where J and bk (λ) are given by (34) and (38).
The function Sn (λ) (41) is expressed in terms of the functions (44) by formula (43). Moreover, Sn (λ) has
the J-properties (36) and factorization (49) takes place, where the factors ak (λ) are given by (47) and (50).
In the second part of the present paper, we plan to establish the relationships between the proposed structures
and de-Branges spaces and nonself-adjoint operators.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, 3rd ed., Pitman, Boston etc. (1981), Vols 1, 2.
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd (1965).
Yu. M. Berezansky, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kyiv (1965).
Yu. Arlinskiĭ and L. Klotz, “Weyl functions of bounded quasiselfadjoint operators and block operator Jacobi matrices,” Acta Sci.
Math. (Szeged), 70, No. 3, 4, 585–626 (2010).
Yu. Arlinskiĭ, “Truncated Hamburger moment problem for an operator measure with compact support,” Math. Nachr., 285, No. 14,
15, 1677–1695 (2012).
D. Damanik, A. Pushnitskii, and B. Simon, “The analytical theory of matrix orthogonal polynomials,” Surv. Approx. Theory, 4, 1–85
(2008).
P. Lopez-Rodriguez, “The Nevanlinna parametrization for a matrix moment problem,” Math. Scand., 89, 245–267 (2001).
P. Lopez-Rodriguez, “Riesz’s theorem for orthogonal matrix polynomials,” Constr. Approx., 15, No. 1, 135–151 (1999).
V. P. Potapov, “The multiplicative structure of J-contractive matrix functions,” Tr. Mosk. Mat. Obshch., 4, 125–236 (1955).
V. A. Zolotarev, Analytic Methods of Spectral Representations of Nonself-Adjoint and Nonunitary Operators [in Russian], KhNU
Publi. House, Kharkov (2003).
M. M. Malamud and S. M. Malamud, “Spectral theory of operator measures in Hilbert space,” St. Petersburg. Math. J., 15, No. 3,
323–373 (2004).
L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, London (1968).
Yu. M. Dyukarev, “Deficiency numbers of symmetric operators generated by block Jacobi matrices,” Sb. Math., 197, No. 8,
1177–1204 (2006).
H. Woracek, “De Branges spaces and growth aspects,” Operator Theory, Springer, Basel (2015).
R. Romanov, “Jacobi matrices and de Branges spaces,” Operator Theory, Springer, Basel (2014).
