Geodesic neighborhoods in unitary orbits of self-adjoint operators of K+C

TL;DR

This paper investigates the geometry of unitary orbits of certain self-adjoint operators, establishing metric geodesics, a local Hopf-Rinow theorem, and analyzing the uniqueness and parametrization of shortest curves.

Contribution

It provides a detailed geometric analysis of unitary orbits in the context of the unitization of compact operators, including new results on geodesics and curve parametrization.

Findings

Established a local Hopf-Rinow theorem for these orbits.

Identified conditions under which shortest curves are unique.

Showed existence of shortest curves not parametrizable by minimal anti-Hermitian operators.

Abstract

We study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group U_(K+C) of the unitization of the compact operators K(H)+C, or equivalently, the quotient U_(K+C)/ U_Diag(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of K(H)+C.