On a conjecture of the norm Schwarz inequality

Abstract

Let $A$ be a positive invertible matrix, and let $B$ be a normal matrix. Following the investigation of Ando, we show that $\Vert A♯\left(B^{\ast }{A}^{-1}B\right)\Vert \ge \Vert B\Vert$, where $♯$ denotes the geometric mean, fails in general.