On the conjecture of the norm Schwarz inequality

Abstract

For any positive invertible matrix $A$ and any normal matrix $B$ in $M_n(\mathbb{C})$, we investigate whether the inequality $||A♯(B^{\ast }{A}^{-1}B)||\ge ||B||$ is true or not, where $♯$ denotes the geometric mean and $||\cdot ||$ denotes the operator norm. We will solve this problem negatively. The related topics are also discussed.

arXiv comment9 pages