Extending positive definiteness

TL;DR

This paper provides criteria for extending sesquilinear form-valued mappings to positive definite forms, leading to new solutions for various truncated moment problems and characterizations of subnormality and unitary dilations.

Contribution

It introduces new criteria for extendibility of sesquilinear forms, enabling solutions to several truncated moment problems and related operator theory characterizations.

Findings

New criteria for extendibility of sesquilinear forms.

Solutions to truncated complex and multidimensional moment problems.

Characterizations of unbounded subnormality and unitary power dilation.

Abstract

The main result of the paper gives criteria for extendibility of sesquilinear form-valued mappings defined on symmetric subsets of *-semigroups to positive definite ones. By specifying this we obtain new solutions of: * the truncated complex moment problem, * the truncated multidimensional trigonometric moment problem, * the truncated two-sided complex moment problem, as well as characterizations of unbounded subnormality and criteria for the existence of unitary power dilation.