Geometrical significance of the lowner-heinz inequality

Abstract

It is proven that the Lowner-Heinz inequality ∥A^tB^t∥ ∥AB∥^t, valid for all positive invertible operators A,B on the Hilbert space H and t ∈ [0, 1], has equivalent forms related to the Finsler structure of the space of positive invertible elements of L(H) or, more generally, of a unital C- algebra. In particular, the Lowner-Heinz inequality is equivalent to some type of "nonpositive curvature" property of that space.