Positive-Definite Operator-Valued Kernels and Integral Representations ()
1. Introduction
About the scalar complex trigonometric moment problem we recall that: a sequence 
of complex numbers with 
is called positive semi-definite if for each
, the Toeplitz matrix 
is positive semi-definite. The problem of characterising the positive semi-definiteness of a sequence of complex numbers was completely solved by Carathéodory in [1], in the following theorem:
Theorem 1. The Toeplitz matrix 
is positive semi-definite and rank 
with 
if and only if the matrix 
is invertible and there exists 
with 
for 
and
such that
(1.1)
In the same paper [1], Charathéodory also proved that: if
, then 
are the roots of the polynomial
which are all distinct and belong to 
Another characterization of the positive semi-definiteness of a sequence of complex numbers was obtained by Herglotz in [2]. In [2], for
, the 
moment of a finite measure 
on 
is defined by
The following characterization of the positivity of a complex moment sequence is the main result in [2].
Theorem 2. A sequence of complex numbers
, 
is positive semi-definite if and only if there exists a positive measure 
on the unit circle 
such that 
From Theorem 1 and Theorem 2, Charathéodory and Fejér in [3] deduce the following theorem:
Theorem 3. Let 
be given complex numbers.
Then there exists a positive measure 
on 
such that
(1.2)
if and only if the Toeplitz matrix 
is positive semi-definite. Moreover, if 
then there exists a positive measure 
supported on 
points of the unit circle 
which satisfies (1.2.)
Theorem 3 gives an answer to the scalar, truncated trigonometric moment problem.
Operator-valued truncated moment problems were studied in [4,5]. Regarding the truncated, trigonometric operator-valued moment problem, we recall that:
1)
, 
is called a spectral function if 
each 
is a bounded, positive operator,
; it is orthogonal if each 
is an othogonal projection;
2) a finite sequence 
of bounded operators on an arbitrary Hilbert space is called a trigonometric moment sequence if, there exists a spectral function 
such that 
for every 
In [4], the necessary and sufficient condition of representing a finite sequence of bounded operators on an arbitrary Hilbert space H, 
with 
as a trigonometric moment sequence is the positivity of the Toeplitz matrix
obtained with the given operators. The representing spectral function is obtained in [4] by generating an unitary operator, defined on the direct sum of 
copies of the Hilbert space H for obtaining an orthogonal spectral function and by applying Naimark’s dilation theorem to get the representing spectral function from it. In [5], a multidimensional operator-valued truncated moment problem is solved. That is: given a sequence of bounded operators
acting on an arbitrary Hilbert space H, with
a necessary and sufficient condition for representing any such operator
as the 
moment of a positive operator-valued measure is given. The necessary and sufficient condition in [5] for such a representation is again the positivity of the Toeplitz matrix
obtained with the given operators. The representing positive operator-valued measure, (spectral function), in [5] is obtained by applying Kolmogorov’s decomposition positive kernels theorem.
Concerning the complex, operator-valued moment problem on a compact semialgebraic nonvoid set
, we recall that a sequence of bounded operators
acting on an arbitrary complex Hilbert spacea H, subject on the conditions
, 
is called a 
moment sequence if there exists an operator-valued positive measure 
on 
such that
A sequence of bounded operators 
with 
and
, acting on an arbitrary, complex, Hilbert space is called a trigonometric operator-valued moment sequence, if there exists a positive, operator-valued measure 
on the p-dimensional complex torus 
such that 
for all
Some of the papers devoted to operator-valued moment problems are: [6-10], to quote only few of them. The operator-valued multidimensional complex moment problem is solved in [9] in the class of commuting multioperators that admit normal extension (subnormal operators) (Theorem 1.4.8., p. 188). In [9], Corollary 1.4.10., a necessary and sufficient condition for solving a trigonometric operator-valued moment problem is given. In [10], another proof of a quite similar necessary and sufficient existence condition on a sequence of bounded operators to admit an integral representation as trigonometric moment sequence with respect to some positive operator valued measure is given. In Section 4 of this note, we prove that the two existence conditions in [9,10] are equivalent.
The present note studies in Section 3 the representation measure of the truncated operator-valued moment problem in [5], only when the given operators act on a finite dimensional Hilbert space. In Proposition 3.1, Section 3, it is shown that the representing measure, in this case, is an atomic one. In Proposition 3.2, Section 3, the necessary and sufficient existence condition in Proposition 3.1 is stated also in terms of matrices.
In Section 4 of the note, is studied the connection between the problem of representing the terms of an operator sequence
as moments of an operator valued, positive measure and the problem of Riesz-Herglotz type integral representation of some operator-valued, analytic function, with positive real part in the class of operators.
2. Preliminaries
Let 
arbitrary,
denote the complex, respectively the real variable in the complex, respectively real euclidian space. For
we denote
and by
. The sets:
represent the torus in 
and 
the unit disc in 
if
and
For
, we denote with 
the integer part of the number 
The addition and subtraction in
, respectively in 
are considered on components. In the set 
the elements are treated in lexicographical order. If 
is an arbitrary complex Hilbert space and
a commuting multioperator, we denote by
for all 
and, as usual, 
is the algebra of bounded operators on
; also 
denotes the Kronecker symbol for
. Let
be a sequence of bounded operators on 
subject to the conditions 
for all
and 
For such a finite sequence of operators, in [5], a necessary and sufficient condition for the existance of a a positive Borel operator-valued measure 
on
, such that the representations
(2.1.)
hold, it is given. Such a measure is called a representing measure for 
In Section 3 of this note, in Proposition 3.1, we give a necessary and sufficient condition for the existence of an atomic representing measure of a truncated, operator-valued moment problem as in (2.1.) in case that the operators 
act on a finite dimensional Hilbert space. In Proposition 3.2 of this note, the necessary and sufficient existence condition for the representing measure in (2.1.) is reformulated in terms of matrices.
In section 4, Proposition 4.2, we establish a RieszHerglotz formula for representing an analytic, operatorvalued function on
, with real positive part in the class of operators. The obtained, representation formula for such functions is the same as in the scalar case [11, 12]. In this case, the representing measure is a positive operator-valued measure. The proof of Proposition 4.1 in this note is based on the characterization on an operatorsequence 
to be a trigonometric, operator-valued moment sequence in [9]. The represented analytic, operator-valued function is the function which has as the Taylor’ s coefficients the operators
.
3. An Operator-Valued Truncated Trigonometric Moment Problem on Finite Dimensional Spaces
Let 
be arbitrary and consider the set
with the lexicographical order (
represents the cartesian product of the mentioned sets), H a finite dimensional Hilbert space with
and 
Proposition 3.1. Let
be a sequence of bounded operators on 
with
for all 
The following assertions are equivalent:
(i) 
for all sequences 
in 
(ii) There exists the multisequence
of 
points and the bounded, positive operators, 
such that
(3.1)
for all 
(iii) There exists a positive atomic operator-valued measure 
on 
such that:
Proof. 
On the set
we have the lexicographical order. The finite sequence of operators 
is considered double indexed i.e.
; with this assumption, from
, 
can be viewed as an operator-valued kernel
Let 
the C-vector space of functions defined on 
with values in the finite dimensional Hilbert space H. With the aid of
, we can introduce on 
the non-negative hermitian product:
according to
, we have the positivity condition:
The matrix associated to this kernel is a Toeplitz matrix of the form:
From Kolmogorov’s theorem, there exists the Hilbert space (essentially unique)
, obtained as the separate completeness of the 
vector space of functions 
with respect to the usual norm generated on the set of cosets of Cauchy sequences, (i.e.
), by the nonnegative kernel
, respectively the space 
(when H is finite dimensional, the Hilbert space
). From the same theorem, there also exists the sequence of operators 
such that 
for all 
In this particular case for
, we have
where 
denotes the range of the operators 
and 
denotes the closed linear span of the sets
,
. The operators 
are:
with 
and 
the Kronecker symbol. Also, from the construction of
, we have
, where 
denotes the range of the operators 
and 
denotes the closed linear span of the sets
.
Let us consider the subsets
the subspaces in
, 
, 
and the operators 
defined by the formula
for any 
with 
the standard basis in
. From the definition of
, since 
are linear for all
, the same is true for the operators 
for all
. For an arbitrary
we have:
for all
. We extend 
to 
preserving the above definition and boundedness condition; the extensions 
are denoted with the same letter 
In case that
are C-linear independent operators with respect to the kernel
, and from above, the operators 
are partial isometries, defined on linear closed subspaces 
with values in
, with equal deficiency indices. In this case, 
admit an unitary extension on the whole space 
for all 
Let us denote the extensions of these operators to 
with the same letter
. The adjoints of 
are defined by
for all 
Obviously, for the extended operators 
In the same time, 
for all 
and all
; we preserve the commuting relations for the extended operators. When 
is a finite dimensional Hilbert space with a basis
, the same is true for the obtained Hilbert space 
All the vectors 
are C-linear independent in 
with respect to the kernel 
Indeed, if
equivalent with
, this equality implies 
We consider that all the vectors 
are C-linear independent in 
with respect to the kernel 
We have then,
.
A basis in 
is
Let 
be the defined isometries, with
and
;
for
and
We have 
and also 
We consider 
the orthonormal algebraic complement of the space 
in
, respectively 
the orthonormal complement of 
When
for 
and 
when
; we have
Let 
be an orthonormal basis in
respectively 
an orthonormal basis in 
We extend the partial isometries 
to the whole spaces 
in the following way:
Because
and
it results that also the extensions are isometries and
; that is 
are unitary operators for all
; ( the extended operators are denoted with the same letters). The commuting relations 
are also preserved 
In the above conditions, the commuting multioperator 
consisting of unitary operators on 
admits joint spectral measurewhose joint spectrum 
Considering the construction of
, we obtain 
and by induction 
for all
Because on the finite dimensional space
, all the operators 
are unitary and compact one, their spectrum 
consists only of the 
principal values. The principal values are the roots of the characteristic polynomials associated with the matrix of 
in suitable basis in
, for all 
The characteristic polynomials of 
are all complex variable polynomials of the same degree
with the roots 
Let
, be the family of the spectral projectors associated with the families of the principal values 
that is 
with 
the spectral measures of 
From the definition of
, we have 
for all
and 
Because
we have also
Consequently, for
, we have obtain:
From Kolmogorov’s decomposition theorem for
, we have
with 
positive operators. That is:
(3.2.)
(i.e. assertion
)
Let 
be a positive, atomic operator-valued measure on
. From 
we have:
(i.e. assertion (iii)).
If
and 
is a positive operator-valued measure, we have:
that is 
Proposition 3.1, in case H a finite dimensional space, statements 
implies also a similar, straightforward characterization, as in the scalar case [6]:
Proposition 3.2. When
operators acting on a finite dimensional space 
with
, are as in Proposition 1, the Toeplitz matrix
is positive semidefinite if and only if it can be factorized as 
with
the diagonal matrix
with entries the positive operators
on the principal diagonal.
4. A Riesz-Herglotz Formula for Operator-Valued, Analytic Functions on the Unit Disk
Remark 4.1. Let 
be a sequence of bounded operators, acting on an arbitrary, separable, complex Hilbert space
, such that 
for all 
and 
The following statements are equivalent:
(a) 
for all 
and all sequences of complex numbers 
with only finite nonzero terms.
(b) There exists a positive, operator-valued measure 
on 
such that
.
(c) The operator kernel 
is positive semidefinite on
, that is it satisfies
for all
, all sequences of vectors 
and all 
Proof. (a) 
(b) was solved in [9], Corollary 1.4.10.
(b) 
(c) represents the sufficient condition in Proposition 1, [10].
(c) 
(a). Let 
with 
for an arbitrary 
From (c), it results
that is the operator kernel satisfies
(that is statement (a)).
Because the trigonometric polynomials are uniformly dense in the space of the continuous functions on 
it results that the representing measure of the operator moment sequence is unique.
For the proof of the following Proposition 4.2, we recall some observations.
A bounded monotonic sequence of positive non-negative operators converges in the strong operator topology to a non-negative operator (pp. 233, [11]). Due to this remark, if 
is a continuous, positive operator-valued function on the compact set
, we define the Riemann integral of the function 
with respect to the Lebesgue measure 
The definition are the usual one in the class of positive operators. That is: the limits of the riemannian sums associated to the function
, arbitrary divisions 
of 
and arbitrary intermediar points 
exists (are limits of bounded monotonic sequence of non-negative operators), and from the continuity assumption of 
on the compact set
, are all the same. We denote the common limits, as usual with 
We apply this natural construction in the proof of the following result.
Proposition 4.2. Let 
be an analytic, vectorial function, with values in the set of bounded operators on a complex, separable Hilbert space
. The following statements are equivalent:
(a) 
(b) (Riesz-Herglotz formula) There exists a positive operator-valued measure 
on 
with
and an operator 
such that:
The proof follows quite the similar steps as the proof of the Riesz-Herglotz formula for analytic, scalar functions with real positive part ([11,12].)
Proof. (a) 
(b) Let
be the Taylor expansion of
, 
with
We define 
for all 
In this case , we obtain for all
,
If we consider 
arbitrary and
, the previous equality becomes
As a consequence of the orthogonality of the system of functions 
with respect to the usual scalar product defined on
, from the the previous remark and 
s uniform convergent expansions, for all sequences 
and all 
we obtain:
We normalize this relation by dividing it with 2 and obtain, for
, the following inequalities:
for all sequences 
and all arbitrary
with
In the above conditions from Theorem 1.4.8, [9], there exists a positive operator-valued measure 
on 
such that
For 
and 
we have 
Let the homeomorphism 
and the positive operator-valued measure
Accordingly to this measure we obtain the representations:
and
Assured by the integral representations of the operators 
we have:
is analytic on
, 
and 
For the operator-valued analytic functions on 
we can state the same characterization theorem as in the the scalar case ( Theorem 3.3, [11],) that is:
Theorem 4.3. Let 
be a sequence of bounded operators acting on an arbitrary, separable, complex Hilbert space
, subject to the conditions 
for all
, 
The following statements are equivalent:
(a) There exists an unique, positive, operator-valued measure 
on 
such that:
(b) The Toeplitz matrix 
is positive semidefinite.
(c) There exists an analytic vectorial function 
for all 
and
for some 
with 
(d) There exists a separable, Hilbert space
, an operator 
and an unitary operator
, such that 
and 
Proof. 
was solved in [9], Th.1.4.8., p. 188. We sketch the proof of implication
.
As in above Proposition 4.2, there exists a positive operator-valued measure 
such that 
In this case, for the function
, we have
that is 
is analytic on 
Also from (a), we have:
From the above representation, it results:
(c) 
(a) As the same proof in Proposition 4.2, we have
for arbitrary
. From this inequality, it results that there exist the representations 
with 
a positive operator valued measure on 
([9], Th. 1.3.2), this is (a).
The equivalence,
. From remark 4.1.we have 
((c) from Remark 4.1.). The equivalence 
is the main result in [10], Proposition 1. p. 116. From [10], Proposition 1, (condition (c) in Remark 4.1.) assured the existence of a Hilbert space
, an operator 
and an unitary operator 
such that
, that is (d); (the Hilbert Space
, the unitary operator 
are obtained by applyng Kolmogorov’s decomposition theorem on positive semidefinite kernels.) Conversely 
is immediately.
5. Conclusion
We give a necessary and sufficient condition on a finite sequence of bounded operators, acting on a finite dimensional Hilbert space, to admit an integral representation as complex moment sequence with respect to an atomic, positive, operator-valued measure. We also established a Riesz-Herglotz representation formula for operator-valued, analytic functions on the unit disc, with real positive part in the class of operators.