On the Truncated Matricial Moment Problem. I

TL;DR

This paper addresses the truncated matrix-valued moment problem, proving a finite atomic representation theorem, characterizing positive functionals, and analyzing the structure of representing measures and core sets.

Contribution

It establishes a matricial version of the Richter-Tchakaloff theorem, shows positive functionals are moment functionals, and investigates the structure of atoms and core sets in the problem.

Findings

Proved a finite atomic representing measure exists for each moment functional.

Established that strictly positive linear functionals are moment functionals.

Proved the equality of the set of atoms and the core set for moment functionals.

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