Abstract

Solutions to some operator-valued, unidimensional, Hamburger and Stieltjes moment problems in this paper are given. Necessary and sufficient conditions on some sequences of bounded operators being Hamburger, respectively, Stieltjes operator-valued moment sequences are obtained. The determinateness of the operator-valued Hamburger and Stieltjes moment sequence is studied.

1. Introduction

A function , , is called a spectral function if(a) is a bounded, positive operator,(b), for any ,(c),(c′) and/or in case and/or .The spectral function is called an orthogonal spectral function if every is an orthogonal projection [1, page 322].(1)A sequence of bounded self-adjoint operators, acting on an arbitrary, complex Hilbert space , subject on the condition , is called a Hamburger, unidimensional operator-valued moment sequence, if there exists an orthogonal spectral function , , such that , , or(2)a sequence , , of bounded self-adjoint operators is called a unidimensional operator-valued Hamburger moment sequence, if there exists a positive operator-valued measure , , measure generated by a spectral function, such that , .

A sequence of bounded positive operators is called a Stieltjes unidimensional operator-valued moment sequence, if there exists a positive operator-valued measure , , (generated by a spectral function) such that , . The passage from the integral representation to an integral representation is done, usually, by applying Naimark’s dilation theorem, or modified forms of it as in [1].

In both cases and , the operator-valued measures or are called the representing measures for the sequence . Necessary and sufficient conditions for representing scalar sequences or operator-valued sequences, in one or several variables, as Hamburger or Stieltjes moment sequences with respect to scalar, respectively, operator-valued, positive measures, represent the subject of many outstanding papers such as [14]…, to quote only few of them.

In the present paper, in Section 3, we give a necessary and sufficient condition on a sequence of bounded, self-adjoint operators to be a Hamburger operator-valued, unidimensional moment sequence. In Section 4, we discuss the uniqueness of the representing measures of the operator-valued Hamburger moment sequence both in and forms. In Section 5, we give some necessary and sufficient conditions on a sequence of positive operators to be a Stieltjes operator-valued, unidimensional moment sequence with respect to a positive, operator-valued measure. The positive representing measures in Sections 3 and 5 are obtained by applying Kolmogorov’s theorem of decomposition of the positive definite kernels.

2. Preliminaries

Let denote the real variable in the real Euclidean space; for an arbitrary complex Hilbert space, represents the algebra of bounded operators an ; we denote with , the function for a Hilbert space, represents the set of bounded operators from in . We consider the C-vector space of vectorial functions: , , with finite support, . We define also the convolution as and make the convention:. We have , with finite support.

In Section 3, a necessary and sufficient condition on a sequence of self-adjoint operators to be a Hamburger operator-valued moment sequence is given. In Section 5, we give necessary and sufficient conditions on a sequence of positive operators to be a Stieltjes operator-valued moment sequence. In Section 4, the problem of the uniqueness of the represented measures in Sections 3 and 5 is studied. The representing measures in Sections 3 and 5 are obtained by applying Kolmogorov’s theorem on decomposition of the positive kernels. Classical Kolmogorov’s theorem for the decomposition of positive kernels is as follows:

“Let be a nonnegative-definite function where is an arbitrary set and a Hilbert space, namely, , for any finite number of points and any vectors . In this case, there exists a Hilbert space (essentially unique) and a function such that for any .

We apply this theorem for a particular set and a particular positive-definite operator-valued function to give an integral representation as Hamburger operator-valued moment sequence and Stieltjes operator-valued moment sequence, respectively, to some sequences of self-adjoint and positive operators, respectively.

3. An Operator-Valued Hamburger Moment Sequence Main Result

Let be a sequence of bounded self-adjoint operators, acting on an arbitrary complex, separable Hilbert space; that is, , , for all , , subject on the following conditions: for any finite vectors’ sequence , there exists another vector sequence such that the following two equations are satisfied;

(A) and for any finite vectors’ sequence , there exists another vectors’ sequence such that

(B)

Proposition 1. Let be a sequence of bounded, self-adjoint operators, acting on an arbitrary complex, separable Hilbert space , subject on the conditions: , (A) and (B) satisfied. The following statements are equivalent. (i)We have:for all sequences with finite support. (ii)There exists a positive operator-valued measure (spectral function) defined on such that

Proof. When , , with finite support} is the C-vector space of functions defined on with vectorial values, we consider the kernel as a double indexed, symmetric one: With the aid of , we introduce the Hermitian, square positive functional , . From property (i) of the kernel , as well as from the properties of the scalar product in , satisfies the following conditions.
(10) is C-linear in the first argument.
(20) , for all .
(30) , for all ; and, moreover, being a Hermitian square, positive functional on , satisfies the Cauchy-Buniakovski-Schwarz inequality, respectively:
(40) Also, from the construction of the Hermitian functional and the symmetry of the kernel , (), the functional satisfies the equalities:
(50) With these assumptions, is a seminorm on . Let be the subset in , defined as with . If follows, using the Cauchy-Buniakovski-Schwarz inequality, that if , we have also ; that is, is a vector subspace in . We consider the separated completion space of with respect to ; that is, in this case, the quotient completion space . Obviously, is a Hilbert space with the usual norm in the completion Hilbert space and is a dense subspace of it (i.e., , where is a Cauchy sequence of elements in and ). The Hilbert space is uniquely defined and it is also described as , (the closed linear span of the ranges of the operators , ). From Kolmogorov’s decomposition theorem of positively defined kernels, with the above construction, the decompositions hold for any . Let us consider the densely defined subspace of , , finite , , and the operator , defined by . We prove that is correctly defined. Consequently, we consider the elements and such that and show that . The above equality is the same as the equality: (modulo ). Indeed, from (50), we have where . From the above definition, we have
(60) and also
(70) From and for arbitrary; we infer that is a densely defined, symmetric operator. We prove that has equal deficiency indices in ; consequently ’s Cayley transform is a partial isometry on with values in . Indeed, let , arbitrary, be the ranges in of the operators . We prove that are vector subspaces in . For this request, we consider the elements , in and arbitrary. Let us define the elements:
,
also the elements , in and . It results that ; and . Because is a symmetric operator, it results also that are closed subspaces in . We prove that, in conditions (A) and (B) for the kernel , we have . Indeed, given an arbitrary element we look for an element , such that . For a construction of the elements like the previous one, we have , , . According to condition (A), on the kernel , such an element exists. We have . Conversely, let ; we search for an element with the property that . Consequently, we have to find an element such that , . We prove with these computations that . That is dim ’s Cayley transform has equal deficiency indices and admits a self-adjoint extension . Let be the spectral measure of the self-adjoint operator . Because for all and , it results that , for all and the integral representations , for all , for all . We consider the positive operator-valued measure . With respect to this positive operator-valued measure, we have , for all and all . That is , for all , the required Hamburger moment integral representations.
Conversely. If the terms admit the integral representations , for all , for a positive operator-valued measure on , we have as it is required by (i).

4. About the Uniqueness of the Hamburger Operator-Valued Moment Sequences’ Representations

Let us consider a sequence of bounded operators , , subject on the condition , , , an arbitrary complex Hilbert space. For the sequence , we get two operator-valued integral representing measures (or spectral functions), , that is, for all . The operator-valued measures allow us to define the scalar measures , , respectively, when is arbitrary. With respect to these scalar measures we obtain From [5, page 283], the Hamburger scalar moment problem is indeterminate (the sequence does not uniquely determine the scalar representing measure). It follows that the operator-valued representing measure does not uniquely determine the Hamburger operator-valued moment sequence.

However, under some additional conditions about the operator-valued representing measure, the Stieltjes (Hamburger) operator-valued moment sequence is determined [3, pages 509, 510, 511].

Moreover, if the representing measure is that associated with a self-adjoint extension of a symmetric operator with deficiency indices (0,0), the self-adjoint extension is the canonical closure of the given operator and is defined on the whole space. Indeed, if is symmetric with and , the canonical closure of , it follows that are closed subspaces in ; that is . In this case the canonical closure of is the smallest self-adjoint extension of and is defined on the whole space (as in Section 3 of this paper, Proposition 1). The same arguments are in [4, page 1267, Lemma 2.1].

Proposition 2. (1) Let , , for all , an arbitrary, complex Hilbert space, subject on the conditions , , and two orthogonal spectral functions on , such that Then on .

Proof. Because , and , the existence of the representation with , is the usual one and is unique. The spectral orthogonal measures coincide; that is, . The representing measure is the spectral orthogonal measure associated with the self-adjoint operator . From ’s multiplicative property, it follows that for all . The uniqueness of the integral representations with respect to spectral functions is assured trivially only in case , for all when the representation is possible.

5. Stieltjes Operator-Valued Moment Sequences

A sequence of bounded operators , acting on an arbitrary Hilbert space , is called a Stieltjes operator-valued moment sequence if there exists a positive operator-valued measure on such that

Proposition 3. Let be an operator sequence with for all and , with conditions (A) and (B) in Proposition 1 satisfied. The following assertions are equivalent.
for all sequences with finite support.
There exists a positive operator-valued measure such that

Proof. We prove that condition [(1) and (2)] is sufficient.
Condition , (1) is the same as (i) in Proposition 1. Consequently, there exists a positive operator-valued measure such that , . In the statement , (2), if we consider the sequence with finite support as , arbitrary, for all , we obtain for all polynomials with complex coefficients and all . It follows that the representing measure is concentrated on .
Conversely. The implication , (1) is the same as from Proposition 1. Moreover, from it results that, for any sequence with finite support, we have that is, , (2).
We give another second characterization on an operator sequence to be an operator-valued Stieltjes moment sequence.

Remark 4. In the sequel, we argue like in [1, page 329]. Between the Hamburger operator-valued moment sequences and Stieltjes moment sequences we can establish the following bijection.
(A) If is a Stieltjes moment sequence with respect to the spectral measure on for the homeomorphism , , there corresponds a spectral measure on defined by such that .
For the homeomorphism , , there corresponds a spectral measure on defined by such that .
We define For we have the representations: , and , .
(B) Conversely. If we have the operator-valued Hamburger moment sequence with respect to the spectral representing measure , respectively, the sequence , defined by for all , admits the integral representation and . We can construct a spectral measure on ; that is, for , we have with , .
For obtaining positive operator-valued measures from orthogonal, spectral functions, in both cases, we compose the orthogonal spectral measure associated with a self-adjoin operator with the same projections, respectively, , in our case. Summing the above conditions (A) and (B) in Remark 4, for an operator-valued sequence , we construct the operator-valued sequence for all and, conversely, for a sequence , we construct the operator-valued sequence , with , .
With the above construction, we have the following.

Proposition 5. The sequence that satisfies conditions (A) and (B) in Proposition 1 is a Stieltjes operator-valued moment sequence if and only if for all sequences with finite support and all .

Proposition (reformulated). The sequence is a Stieltjes operator-valued moment sequence if and only iffor all sequences with finite support and all , , for all defined above.

Proof. Let be an operators’ sequence, , an arbitrary complex Hilbert space; we define , ; that is, and , for all . In the condition of the hypothesis (Propositions 5 and 5′), we have for all sequences with finite support. From Proposition 1, there exists a positive operator-valued measure on such that From Remark 4, there exists a positive operator-valued measure such that That is is a Stieltjes operator-valued moment sequence.
Conversely. If , , we construct a measure positively defined on , as in Remark 4, with the property that ; and . In this case,

Proposition 6. Let be an operator sequence with for all and . The following assertions are equivalent:
for all sequences with finite support.
There exists a positive operator-valued measure such that

Proof. We prove that condition [ and ] is sufficient.
Condition , is the same with (i) in Proposition 1. Consequently, there exists a positive operator-valued measure such that , . In the statement , , if we consider the sequence with finite support as , , arbitrary, for all , we obtain for all polynomials with complex coefficients and all . It follows that the representing measure is concentrated on .
Conversely. The implication , (1) is the same with from Proposition 1. Moreover, from it results that, for any sequence with finite support, we have that is , (2).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.