Inequalities for Unitarily Invariant Norms

Abstract

Let A,B,X be complex matrices with A,B positive semidefinite. It is proved that (2+t)||Ar XB2-r+A2-rXBr ||\le 2||A2X +tAXB+XB2 || for any unitarily invariant norm $||\cdot||$ and real numbers r,t satisfying $1\le 2r\le 3,$ $-2 < t\le 2.$ The case r=1, t=0$ of this result is the well-known arithmetic-geometric mean inequality due to R. Bhatia and C. Davis [SIAM J. Matrix Anal. Appl., 14 (1993), pp. 132--136]. Several other unitarily invariant norm inequalities are derived.