On Symmetric Norm Inequalities And Hermitian Block-Matrices

The main purpose of this paper is to englobe some new and known types of Hermitian block-matrices $M=\left(\begin{array}{c}A\mathrm{\&}X\\ X^{\ast }\mathrm{\&}B\end{array}\right)$ satisfying or not the inequality $\mathrm{\parallel }M\mathrm{\parallel }\le \mathrm{\parallel }A+B\mathrm{\parallel }$ for all symmetric norms