On the Truncated Matricial Moment Problem. I

Abstract

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 $\mathcal{W}(\Lambda)=\mathcal{V}(\Lambda)$.