On the conjecture of the norm Schwarz inequality

TL;DR

This paper investigates a conjecture involving the operator norm of the geometric mean of matrices and provides a negative answer, contributing to the understanding of matrix inequalities.

Contribution

It resolves a conjecture about the inequality involving the geometric mean of matrices, showing that the proposed inequality does not hold in general.

Findings

The inequality $||Alacklozenge (B^{*}A^{-1}B)|| geq ||B||$ is false in general.

The paper discusses related topics in matrix inequalities and operator norms.

It clarifies the limitations of the conjectured inequality in matrix analysis.

Abstract

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