This paper is about the general truncated matrix-valued moment problem. Let $\mathcal{ H}_q$ denote the complex Hermitian $q\times q$-matrices, $q\in \mathbb{ N}$. Suppose that $(\mathcal{ X},\mathfrak{ X})$ is a measurable space and $\mathcal{ E}$ is a finite-dimensional vector space of measurable mappings of $\mathcal{ X}$ into $\mathcal{ H}_q$. A linear functional $\Lambda$ on $\mathcal{ E}$ is called a moment functional if there exists a positive $\mathcal{ H}_q$-valued measure $\mu$ on $(\mathcal{ X},\mathfrak{ X})$ such that $\Lambda(F)=\int_\mathcal{ X} \langle F,\mathrm{ d}\mu\rangle$ for $F\in \mathcal{ E}$. We prove a matricial version of the Richter-Tchakaloff theorem which states that each moment functional on $\mathcal{ E}$ has a finitely atomic representing measure. It is shown that strictly positive linear functionals on $\mathcal{ E}$ are moment functionals. For a moment functional $\Lambda$, we study the set of atoms $\mathcal{ W}(\Lambda)$ and the Carath\'eodory numbers $\mathrm{ Car}(\Lambda)$, $\mathrm{ car}(\Lambda)$ and we define and investigate the core set $\mathcal{ V}(\Lambda)$. A main result of the paper is the equality $\mathcal{ W}(\Lambda)=\mathcal{ V}(\Lambda)$.
Categories: math.FA math.AG