Some norm inequalities for some positive block matrices

Antoine Mhanna;


We review Lin's inequality from \cite{ML} and (re)prove that if $M=[X_{i,j}]_{i,j=1}^n$ is positive semi-definite then $\bigoplus_{i=1}^{n}(X_{i,i}-\sum\limits_{j\neq i}X_{j,j})\le M^\tau \le I_n \otimes \sum\limits_{i=1}^nX_{i,i}$ where $M^{\tau}$ is the partial transpose of $M$. In particular for such $M\in {\mathbb{M}}_n^{{+}}({\mathbb{M}}_m)$ we prove that $\|M\|\le \min(m,n)\| \sum\limits_{i=1}^nX_{i,i}\|$ for all symmetric norms. Some classical results are also discussed in terms of permutations.

Vol. 24 (2023), No. 1, pp. 301-308

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