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

A Real Nullstellensatz for free modules

“…(3) and (4) have already been solved in [1,Theorems 14 and 18] by using the ideas from [33,Section 4]. In special cases, however, stronger results exist; see [18,20,3,4] [19,10,9,28] if d = 1 and [17,21,35] if G i and F are linear.…”

Bounding the separable rank via polynomial optimization

“…In this section we prove the implication (ii) =⇒ (i) in Theorem 9 from its real analog in [10], which we restate below for convenience. x Im ] → S 2m defined via (90).…”

Non-negative Spectral Measures and Representations of C*-Algebras

“…Section 4 provides a characterization of nonnegative measures (see Theorem 4). In Section 5 we extend the integration theory to a Banach space which in particular constists of all bounded measurable W 1 -valued functions and obtain a slight extension of [3,Proposition 2]; see Theorem 5. In Section 6 we show how our measures are connected with the measures from [8] (see Proposition 4).…”

“…In this paper we show that for every compact Hausdorff space X and every von Neumann algebras W 1 , W 2 there is a one-to-one correspondence between unital * -representations ρ : C(X, W 1 ) → W 2 and special B(W 1 , W 2 )-valued measures on X that we call non-negative spectral measures. Such measures are special cases of non-negative measures that we introduced in our previous paper [3] in connection with moment problems for operator polynomials. …”