On a criterion for the determinate-indeterminate dichotomy of the moment problem

TL;DR

This paper introduces a new criterion for classifying the Hamburger moment problem as determinate or indeterminate based on the structure of bases for Jacobi operators, linking operator selfadjointness to solution uniqueness.

Contribution

It provides a novel criterion using bases of representation for Jacobi operators, enhancing understanding of the moment problem's determinacy.

Findings

New criterion for moment problem classification

Connection between operator bases and solution uniqueness

Structural conditions on Jacobi matrices

Abstract

When the classical Hamburger moment problem has solutions, it has either exactly one solution or infinitely many solutions. Correspondingly, the moment problem is said to be either determinate or indeterminate. In terms of Jacobi operators, this dichotomy translates into the operator being either selfadjoint or symmetric nonselfadjoint. In this work, we present a new criterion for the determinate-indeterminate classification which hinges on bases of representation (in Akhiezer-Glazman terminology) for Jacobi operators so that the corresponding matrices have a certain structure.