On the Truncated Matricial Moment Problem. I

2023/10/02 by Conrad Mädler, Konrad Schmüdgen, Mädler, Conrad +1 · 4 citations
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Advanced Topics in Algebra #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2310.00957

Abstract

This paper is about the general truncated matrix-valued moment problem. Let H_q denote the complex Hermitian q× q-matrices, q∈ ℕ. Suppose that (X,\mathfrakX) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into H_q. A linear functional Λ on E is called a moment functional if there exists a positive H_q-valued measure μ on (X,\mathfrakX) 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 the Carathéodory numbers Car(Λ), car(Λ) and we define and investigate the core set V(Λ). A main result of the paper is the equality W(Λ)=V(Λ).

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