Self-adjoint operators associated with Hankel moment matrices

TL;DR

This paper characterizes self-adjoint operators linked to Hankel moment matrices, extending previous work to cases with indeterminate moments and finite index of determinacy, using moment-based proofs.

Contribution

It provides a new proof for the closure of Hankel forms and describes associated self-adjoint Hankel operators in broader cases beyond Yafaev's original work.

Findings

Describes self-adjoint Hankel operators for various moment conditions.

Provides a moment-based proof for the closure of Hankel forms.

Extends analysis to indeterminate and finite index cases.

Abstract

In a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments $(m_n)$ of a positive measure with infinite support on the real line. If $m_n=o(1)$ Yafaev characterized the closure of the form based on earlier work on quasi-Carleman operators. We give a new proof of the description of the closure based entirely on moment considerations. The main purpose of the present paper is a description of the self-adjoint Hankel operators associated with closed Hankel forms in the Hilbert space of square summable sequences. We do this not only in the case $m_n=o(1)$ studied by Yafaev but also in two other cases, where the Hankel form is closable, namely if the moment sequence is indeterminate or if the moment sequence is determinate with finite index of determinacy.