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 antiHermitian bounded diagonal operators in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators Uk,d = {u ∈ U(H) : ∃D ∈ D(B(H) ah), u − e D ∈ K(H)} in order to obtain a concrete description of short curves in unitary Fredholm orbits Ob = {e Kbe−K : K ∈ 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 ∈ Ob and x ∈ Tc(Ob) there exists a minimal lifting Z0 ∈ B(H) ah (in the quotient norm, not necessarily compact) such that γ(t) = e tZ0 ce−tZ0 is a short curve on Ob in a certain interval.