Differential Geometry for Nuclear Positive Operators

CONDE, CRISTIAN

INTEGRAL EQUATIONS AND OPERATOR THEORY

Birkhäuser Basel

Año: 2007 vol. 57 p. 451 - 471

0378-620X

Abstract.  Let H be a Hilbert space, . The set Ä1 = {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 Ä1, 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.