Unitary orbits in a full matrix algebra

TL;DR

This paper explores the geometric and topological properties of unitary orbits within the manifold of positive invertible Hilbert-Schmidt operators, focusing on smooth structures and convexity under different Riemannian metrics.

Contribution

It provides conditions for smooth local structures of unitary orbits and analyzes their convexity in the context of Riemannian geometry on the manifold.

Findings

Conditions for smooth local structure of orbits

Convexity properties under different metrics

Insights into the geometry of positive invertible operators

Abstract

The Hilbert manifold $\Sigma$ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits $\Omega\subset \Sigma$ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set $\Omega$, and we study the convexity of this set for the geodesic structures that arise when we give $\Sigma$ two Riemannian metrics.