Moment problems for 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.