Minimal length curves in unitary orbits of a Hermitian compact operator

TL;DR

This paper investigates minimal length curves in the unitary orbits of Hermitian compact operators, exploring their properties, approximations, and relation to anti-Hermitian operators within the framework of homogeneous spaces.

Contribution

It introduces new examples of minimal curves not characterized by anti-Hermitian operators but can be approximated by such curves, expanding understanding of geometric structures in operator spaces.

Findings

Existence of minimal curves not of anti-Hermitian type

Approximation of these curves by anti-Hermitian-based minimal curves

Insights into geometric properties of unitary orbits in B(H)

Abstract

We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.