Finsler geometry and actions of the $p$-Schatten unitary groups

Abstract

Let p be an even positive integer and U p (H) the Banach-Lie group of unitary operators u which verify that u - 1 belongs to the p-Schatten ideal B p (H). Let O be a smooth manifold on which U p (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 U p (H). Our main result establishes that for any pair of given initial conditions x ∈O and X ∈ (TO) x there exists a curve λ(t) = e tz • x in O,with z a skew-hermitian element in the p-Schatten class, such that λ(0) = x and λ(0) = X, which remains minimal as long as t∥z∥p ≤ n/4. Moreover, λ 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 U p (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 ∈ U p (H)} of a self-adjoint operator A ∈B(H). © 2009 American Mathematical Society.