An analytical parameterization for all solutions of the two-dimensional moment problem under Carleman-type conditions

TL;DR

This paper provides an explicit parameterization of all solutions to the two-dimensional moment problem under Carleman-type conditions, using analytic operator-valued functions and generalized resolvents of symmetric operators.

Contribution

It introduces a comprehensive analytical parameterization for solutions of the 2D moment problem under specific Carleman conditions, including the determinate case.

Findings

All solutions are parameterized by analytic contractive operator-valued functions.

The determinate case of the moment problem is characterized.

A notion of generalized resolvent for commuting symmetric operators is introduced.

Abstract

The two-dimensional moment problem consists of finding a positive Borel measure $\mu$ in $\mathbb{R}^2$ such that $\int_{\mathbb{R}^2} t_1^m t_2^n d\mu = s_{m,n}$, $m,n=0,1,2,...$, where $s_{m,n}$ are prescribed real constants (moments). We study this moment problem in the case when the sequence $\{ s_{m,n} \}_{m,n=0}^\infty$ is positive semi-definite, and the following Carleman-type conditions hold: $$ \sum_{k=1}^\infty \frac{1}{ \sqrt[2k]{ s_{2m,2k} + s_{2m+2,2k} } } = \infty,\quad m=0,1,2,.... $$ In this case all solutions of the moment problem are parameterized by a class of analytic contractive operator-valued functions. The special case of the determinate moment problem is characterized. We introduce a notion of a generalized resolvent for a pair of commuting symmetric operators. We use basic properties of such generalized resolvents as a main tool in studying the above moment problem.