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 Schmüdgen to matrix polynomials and show with a counterexample, that the extension to the operator polynomials fails in general.

References in corpus (4)

• Infinite J-matrices and a matrix moment problem
• A description of all solutions of the matrix Hamburger moment problem in a general case
• Multivariate Determinateness
• The matrix Stieltjes moment problem: a description of all solutions

Cited by in corpus (13)

• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• A Real Nullstellensatz for Free Modules
• The strong truncated Hamburger moment problem with and without gaps
• Finsler's Lemma for Matrix Polynomials
• Some Positivstellensätze for polynomial matrices
• The singular bivariate quartic tracial moment problem
• The Local Operator Moment Problem on $\mathbb{R}$
• A Moment-SOS Hierarchy for Robust Polynomial Matrix Inequality Optimization with SOS-Convexity
• Matrix Fejér-Riesz theorem with gaps
• Handelman's Positivstellensatz for polynomial matrices positive definite on polyhedra
• Some applications of Scherer-Hol's theorem for polynomial matrices
• Integral representations of *-representations of *-algebras
• Non-negative spectral measures and representations of C*-algebras