Geometrical significance of the Löwner-Heinz inequality

Abstract

It is proven that the Löwner-Heinz inequality ‖ A t B t ‖ ≤ ‖ A B ‖ t {\|A^{t}B^{t}\|\le \|AB\|^{t}} , valid for all positive invertible operators A , B {A, B} on the Hilbert space H {\mathcal H } and t ∈ [ 0 , 1 ] {t\in [0,1]} , has equivalent forms related to the Finsler structure of the space of positive invertible elements of L ( H ) {\mathcal L (\mathcal H )} or, more generally, of a unital C ∗ {C^{*}} -algebra. In particular, the Löwner-Heinz inequality is equivalent to some type of “nonpositive curvature" property of that space.