Finsler geometry and actions of the p-Schatten unitary groups

Abstract

Let p be an even positive integer and Up(H) the Banach-Lie group of unitary operators u which verify that u-1 belongs to the p-Schatten ideal Bp(H). Let O be a smooth manifold on which Up(H) acts transitively and smoothly. Then one can endow O with a natural Finsler metric in terms of the p-Schatten norm and the action of Up(H). Our main result establishes that for any pair of given initial conditions X∈O and ∈ (TO) x there existsa curve δ(t)=etz . x in O, with z a skew-hermitian element in the p-Schatten class, such that δ(0)=xand δ(0)=X, which remains minimal as long as tkzkp≤ π/4. Moreover, δ is unique with these properties. We also show that the metric space (O, d) (where d is the rectifiable distance) is complete. In the process we establish minimality results in the groups Up(H), and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit O = {uAu∗ : u ∈ Up(H)} of a self-adjoint operator A ∈ B(H).