Unitary subgroups and orbits of compact self-adjoint operators

VARELA, ALEJANDRO; BOTTAZZI, TAMARA

STUDIA MATHEMATICA

POLISH ACAD SCIENCES INST MATHEMATICS

Lugar: VARSOVIA; Año: 2017 vol. 238

0039-3223

Let H be a separable Hilbert space, and D(B(H)ah) theanti-Hermitian bounded diagonals in some fixed orthonormal basis andK(H) the compact operators. We study the group of unitary operators Uk,d = {u ∈ U(H) : ∃ D ∈ D (B (H))ah such that u − e^D ∈ K(H)}in order to obtain a concrete description of short curves in unitaryFredholm orbits of a compact selfadjointoperator b with spectral multiplicity one. We consider the rectifiabledistance on the orbit defined as the infimum of curve lengths measuredwith the Finsler metric defined by means of the quotient spaceK(H)^ah/D(K(H)^ah). Then for every c in the orbit and x in its tangent space there exist a minimal lifting Z0 ∈ B(H)^ah (in the quotient norm, not necessarily compact) such that γ(t) = e^(tZ0) c e^(−tZ0)is a short curve on the orbit in a certain interval.