Moment problems for operator polynomials

Haviland's theorem states that given a closed subset K in Rn, each functional L:R[x¯]→R positive on Pos(K)≔{p∈R[x¯]|p|K≥0} admits an integral representation by a positive Borel measure. Schmüdgen proved that in the case of compact semialgebraic set K it suffices to check positivity of L on a preordering T, having K as the non-negativity set. Further he showed that the compactness of K is equivalent to the archimedianity of T. The aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into R,Mn(R) or B(K).