# On the structure of Hausdorff moment sequences of complex matrices

## TL;DR

This paper investigates the intrinsic structure of truncated matricial Hausdorff moment sequences, revealing their bounded variation within specific intervals and identifying extremal solutions using algebraic methods and matrix parallel sums.

## Contribution

It characterizes the structure of matricial Hausdorff moment sequences and introduces extremal solutions based on endpoint conditions, employing algebraic techniques and matrix parallel sums.

## Key findings

- Each moment sequence varies within a closed bounded interval.
- Endpoints of the interval correspond to distinguished solutions.
- Parallel sum of matrices is key in the proofs.

## Abstract

The paper treats several aspects of the truncated matricial $[\alpha,\beta]$-Hausdorff type moment problems. It is shown that each $[\alpha,\beta]$-Hausdorff moment sequence has a particular intrinsic structure. More precisely, each element of this sequence varies within a closed bounded matricial interval. The case that the corresponding moment coincides with one of the endpoints of the interval plays a particular important role. This leads to distinguished molecular solutions of the truncated matricial $[\alpha,\beta]$-Hausdorff moment problem, which satisfy some extremality properties. The proofs are mainly of algebraic character. The use of the parallel sum of matrices is an essential tool in the proofs.