Some norm inequalities for some positive block matrices
Résumé
We review Lin's inequality of numerical range widths and 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 $M$. Some classical results are also discussed in terms of permutations.
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