Nonpositive curvature in p-Schatten class

We study the geometry of the set Δ 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 d p and exposed several results about this metric space. The aim of this work is to prove that the space ( Δ p , d p ) behaves in many senses like a nonpositive curvature metric space.