Application of Localization to the Multivariate Moment Problem

It is explained how the localization technique introduced by the author in [19] leads to a useful reformulation of the multivariate moment problem in terms of extension of positive semidefinite linear functionals to positive semidefinite linear functionals on the localization of R[x] at p = ∏n i=1(1 + x 2 i ) or p ′ = ∏n−1 i=1 (1 + x 2 i ). It is explained how this reformulation can be exploited to prove new results concerning existence and uniqueness of the measure μ and density of C[x] in Ls(μ) and, at the same time, to give new proofs of old results of Fuglede [11], Nussbaum [21], Petersen [22] and Schmüdgen [27], results which were proved previously using the theory of strongly commuting self-adjoint operators on Hilbert space.