Moment problems for operator polynomials

TL;DR

This paper extends classical moment problem results to operator polynomials, establishing integral representations for positive linear maps and highlighting limitations with operator polynomial extensions.

Contribution

It generalizes Haviland's theorem and Schmüdgen's results from polynomials to operator polynomials, identifying where these extensions succeed and fail.

Findings

Extended Haviland's theorem to operator polynomials

Extended Schmüdgen's result to matrix polynomials

Provided a counterexample showing failure of extension to operator polynomials

Abstract

We extend Haviland's theorem on the integral representation of positive linear functionals on usual (real multivariate) polynomials to the integral representation of positive linear maps on operator polynomials mapping into the space of operators. We also extend its special case of compact semialgebraic set studied by Schm\"udgen to matrix polynomials and show with a counterexample, that the extension to the operator polynomials fails in general.