Nonpositive Curvature in p-Schatten class

Abstract

We study the geometry of the set p, with 1 < p < ∞, which consists of perturbations of the identity operator by p-Schatten class operators, which are positive and invertible as elements of B(H). These manifolds have natural and invariant Finsler structures. In [C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008) 920– 931], we introduced the metric dp and exposed several results about this metric space. The aim of this work is to prove that the space (p,dp) behaves in many senses like a nonpositive curvature metric space.