The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$

Abstract:

The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$ will be considered. The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$ requires necessary and sufficient conditions for a multisequence of Hermitian matrices $\{ S_{\gamma } \}_{\gamma \in \Gamma }$ (where $\Gamma$ is a finite subset of $\mathbb {N}_0^d$) to be the corresponding moments of a positive Hermitian matrix-valued Borel measure $\sigma$, and also the support of $\sigma$ must be contained in some given non-empty set $K \subseteq \mathbb {R}^d$, i.e., \begin{equation*} S_{\gamma } = \int _{\mathbb {R}^d} \xi ^{\gamma } d\sigma (\xi ),\;\;\textrm {for}\;\textrm {all}\; \gamma \in \Gamma , \tag {0.1}\end{equation*} and \begin{equation*} \textrm {supp} \; \sigma \subseteq K. \tag {0.2}\end{equation*} Given a non-empty set $K \subseteq \mathbb {R}^d$ and a finite multisequence, indexed by a certain family of finite subsets of $\mathbb {N}_0^d$, of Hermitian matrices we obtain necessary and sufficient conditions for the existence of a minimal finitely atomic measure which satisfies (0.1) and (0.2). In particular, our result can handle the case when $\Gamma = \{ \gamma \in \mathbb {N}_0^d \colon 0 \leq |\gamma | \leq 2n+1 \}$. We will also discuss a similar result in the multivariable complex and polytorus setting.