On Symmetric Norm Inequalities And Hermitian Block-Matrices

TL;DR

This paper investigates inequalities involving Hermitian block-matrices and symmetric norms, providing new insights into when certain norm inequalities hold or fail for these matrices.

Contribution

It introduces new conditions and results characterizing when Hermitian block-matrices satisfy specific symmetric norm inequalities.

Findings

Identifies classes of Hermitian block-matrices satisfying the inequality M  A+B for all symmetric norms

Provides counterexamples where the inequality does not hold

Extends known results on matrix norm inequalities for block-structured matrices

Abstract

The main purpose of this paper is to englobe some new and known types of Hermitian block-matrices $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix}$ satisfying or not the inequality $\|M\|\le \|A+B\|$ for all symmetric norms