A Cauchy-Schwarz inequality for operators with applications

Abstract

For any unitarily invariant norm on Hilbert-space operators it is shown that for all operators A, B, X and positive real numbers r we have ||| |A^∗XB|^r |||² ||| |AA^∗X|^r ||| ||| |XBB^∗|^r |||. Some consequences are then discussed. A simple proof is given for the fact that for positive operators A, B the function [spr(A^tB^t)]^1/t is monotone in t on the positive half line.