ISRN Mathematical Analysis

Volume 2014 (2014), Article ID 836839, 7 pages

http://dx.doi.org/10.1155/2014/836839

Research Article

Hamburger and Stieltjes Moment Problems for Operators

L. Lemnete-Ninulescu

Department of Mathematics, “Politehnica” University of Bucharest, Splaiul Independenţei, No. 313, 060042 Bucharest, Romania

Received 2 December 2013; Accepted 8 January 2014; Published 30 April 2014

Academic Editors: D. D. Hai and S. Pilipovic

Copyright © 2014 L. Lemnete-Ninulescu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1–4]…, 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 if

for 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.

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