Hamburger and Stieltjes Moment Problems for Operators

L Lemnete-Ninulescu

doi:10.1155/2014/836839

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.

Highlights

Let t ∈ R denote the real variable in the real Euclidean space; for H an arbitrary complex Hilbert space, L(H) represents the algebra of bounded operators an H; we denote with δi⋅ : N → {0, 1}, the function
There exists a Hilbert space K and a function h : S → B(H, K) such that Γ(s, t) = h(t)∗h(s) for any s, t ∈ S.”. We apply this theorem for a particular set S 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
There exists a positive operator-valued measure EΓ : Bor(R) → A(H) such that Γn = ∫−+∞∞ tndEΓ(t), n = 0, 1, 2,

Summary

Introduction

A function E(λ), a ≤ λ ≤ b, is called a spectral function if (a) E(λ) is a bounded, positive operator,. H, subject on the condition A0 = IdH, is called a Hamburger, unidimensional operator-valued moment sequence, if there exists an orthogonal spectral function E(λ), −∞ ≤ λ ≤ ∫−+∞∞ λndE(t), n = 0, 1, 2, . A sequence {An}+n=∞0 of bounded positive operators is called a Stieltjes unidimensional operator-valued moment sequence, if there exists a positive operator-valued measure F(λ), 0 ≤ λ ≤ +∞, (generated by a spectral function) such that An = ∫0+∞ λndF(λ), n = 0, 1, 2,. The passage from the integral representation (1) to an integral representation (2) is done, usually, by applying Naimark’s dilation theorem, or modified forms of it as in [1] In both cases (1) and (2), the operator-valued measures E(λ) or F(λ) are called the representing measures for the sequence {An}+n=∞0.

Preliminaries

An Operator-Valued Hamburger Moment Sequence Main Result

Stieltjes Operator-Valued Moment Sequences