Moment problems for operator polynomials

Haviland’s theorem states that given a closed subset K in R n , 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 , M n ( R ) or B ( K ) .