Truncated K-moment problems in several variables

Abstract

Let $\beta \equiv {\beta }^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(\beta )$, and let $r:=rank\mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure μ, which is r-atomic, with supp μ$equalto$\mathcal{V}(\mathcal{M}(n+1))$,thealgebraicvarietyof$\mathcal{M}(n+1)$.Further,\beta hasanr-atomic(minimal)representingmeasuresupportedinasemi-algebraicset$K_{\mathcal{Q}}$subordinatetoafamily$\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$ifandonlyif$\mathcal{M}(n)$ispositivesemidefiniteandadmitsarank-preservingextension$\mathcal{M}(n+1)$forwhichtheassociatedlocalizingmatrices$\mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$arepositivesemidefinite$(1\leq i\leq m)$;inthiscase,\mu (asabove)satisfiessupp\mu \subseteq K_{\mathcal{Q}}$, and μhas precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$atomsin$\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$,$1\leq i\leq m$.

33 pages; to appear in J. Operator Theory