Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices

Abstract

For positive semi-definite block-matrix $M,$ we say that $M$ is P.S.D. and we write $M=\left(\begin{array}{c}A\&X\\ X^{\ast }\&B\end{array}\right)\in \mathbb{M}\_{n+m}^{+}$, with $A\in \mathbb{M}\_n^{+}$, $B\in \mathbb{M}\_m^{+}.$ The focus is on studying the consequences of a decomposition lemma due to C.~Bourrin and the main result is extending the class of P.S.D. matrices $M$ written by blocks of same size that satisfies the inequality: $\Vert M\Vert \le \Vert A+B\Vert$ for all symmetric norms.