A Cauchy-Schwarz inequality for operators with applications

AbstractFor 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(AtBt)]1t is monotone in t on the positive half line