Unitary subgroups and orbits of compact self-adjoint operators

Abstract

Let H be a separable Hilbert space, and let D(B(H))^ah be the anti-Hermitian bounded diagonal operators in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators Uk,d={u:∃D∈D(B(H))^ah ,u−e^D∈K(H)} in order to obtain a concrete description of short curves in unitary Fredholm orbits O_b={e^Kbe^{−K}:K∈K(H)^ah} of a compact self-adjoint operator b with spectral multiplicity one. We consider the rectifiable distance on O_b 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∈O_b and x∈Tc(O_b) there exists a minimal lifting Z0∈K(H)^ah (in the quotient norm, not necessarily compact) such that γ(t)=e^{tZ0}ce^{−tZ0} is a short curve on O_b in a certain interval.