On the truncated matricial moment problem. I

This paper is about the general truncated matrix-valued moment problem. Let Hq denote the complex Hermitian q×q-matrices, q∈N. Suppose that (X,X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into Hq. A linear functional Λ on E is called a moment functional if there exists a positive Hq-valued measure μ on (X,X) such that Λ(F)=∫X〈F,dμ〉 for F∈E. We prove a matricial version of the Richter–Tchakaloff theorem which states that each moment functional on E has a finitely atomic representing measure. It is shown that strictly positive linear functionals on E are moment functionals. For a moment functional Λ, we study the set of atoms W(Λ) and define two Carathéodory numbers of Λ. Further, we define and investigate the core set V(Λ) which generalizes the core variety from the scalar case to the matricial setting. A main result of the paper is the equality W(Λ)=V(Λ).