Unitary subgroups and orbits of compact self-adjoint operators

TL;DR

This paper characterizes the geometry of unitary orbits of compact self-adjoint operators on a Hilbert space, providing explicit descriptions of shortest paths using a specialized Finsler metric.

Contribution

It offers a concrete description of short curves in unitary Fredholm orbits of compact self-adjoint operators with spectral multiplicity one, using a quotient space and minimal liftings.

Findings

Explicit description of short curves in unitary orbits

Existence of minimal liftings for tangent vectors

Characterization of rectifiable distances in the orbit space

Abstract

Let H be a separable Hilbert space, and D(B(H))^ah the anti-Hermitian bounded diagonals in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators U_kd = {u in U(H): such that u-e^D in K(H) for D in D(B(H))^ah} in order to obtain a concrete description of short curves in unitary Fredholm orbits Ob={ e^K b e^{-K} : K in K(H)^ah } of a compact self-adjoint operator b with spectral multiplicity one. We consider the rectifiable distance on Ob defined as the infimum of curve lengths measured with the Finsler metric defined by means of the quotient space K(H)^ah / D(K(H)^ah). Then for every c in Ob and x in T(\ob)_c there exist a minimal lifting Z_0 in B(H)^ah (in the quotient norm, not necessarily compact) such that g(t)=e^{t Z_0} c e^{-t Z_0} is a short curve on Ob in a certain interval.