Differential geometry on Thompson's components of positive operators

Abstract

Consider the algebra L(H) of bounded linear operator on a Hilbert space H, a let L(H)^+ be the set of positiveelements of L(H). For each A ∈ L(H)^+ we study differential geometry of the Thompson component of A, C_A={B ∈ L(H)^+ : A ≤ rB and B ≤ sA for some s,r >0}. The set components is parametrized by means of all operator ranges of H. Each C_A is a differential manifold modelled in an appropiate Banach space and a homogeneous space with a natural connection. Morover, given arbitrary B,C ∈ C_A, there exists a unique geodesic with endpoints B and C. Finally, we introduce a Finsler metric on C_A for which the geodesics are short and we show that in coincides with the so-called Thompson metric.