Differential Geometry for Nuclear Positive Operators

Abstract

Let H be a Hilbert space, infinite dimensional. The set Delta_1 = {1 + a : a in the trace class, 1 + a positive and invertible} is a differentiable manifold of operators, and a homogeneous space under the action of the invertible operators g which are themselves nuclear perturbations of the identity (one of the called classical Banach-Lie groups): l_g(1 + a) = g(1 + a)g*. In this paper we introduce a Finsler metric in Delta_1 , which is invariant under the action. We investigate the metric space thus induced. For instance, we prove that it is complete non-positively curved (in the sense of Busemann). Other geometric properties are derived.