Norm inequalities for accretive-dissipative block matrices

Abstract

Let $ T=[T_{ij}]\in \mathbb{M} _{mn}(\mathbb{C})$ be accretive-dissipative, where $T_{ij}\in \mathbb{M} _{n}(\mathbb{C} )$ for $i,j=1,2,...,m.$ Let $f$ be a function that is convex and increasing on $ [0,\infty )$ where $f(0)=0.$ Then $$ \left\vert \left\vert \left\vert f\left(\sum_{i < j}\left\vert T_{ij}\right\vert^{2}\right) +f\left(\sum_{i < j}\left\vert T_{ji}^{\ast}\right\vert^{2}\right) \right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert f\left( \frac{m^{2}-m}{2}\left\vert T\right\vert^{2}\right) \right\vert \right\vert \right\vert. $$ Also, if $f$ is concave and increasing on $[0,\infty )$ where $f(0)=0$, then% \begin{equation*} \left\vert \left\vert \left\vert f\left( \sum\limits_{i < j}\left\vert T_{ij}\right\vert ^{2}\right) +f\left( \sum\limits_{i < j}\left\vert T_{ji}^{\ast }\right\vert ^{2}\right) \right\vert \right\vert \right\vert \leq (2m^{2}-2m)\left\vert \left\vert \left\vert f\left( \frac{\left\vert T\right\vert ^{2}}{4}\right) \right\vert \right\vert \right\vert. \end{equation*}

Нехай $T=T_{ij}\in \mathbb{M}_{mn}(\mathbb{C} )$, де $T_{ij}\in \mathbb{M}_{n}(\mathbb {C})$ при $i,j=1,2,...,m.$, -- акретивно-дисипативна матриця. Нехай $f$ - опукла функція, яка зростає на $ [0,\infty )$, де $f(0)=0.$ Тоді \begin{equation*} \left\vert \left\vert \left\vert f\left( \sum\limits_{i < j}\left\vert T_{ij}\right\vert ^{2}\right) +f\left( \sum\limits_{i < j}\left\vert T_{ji}^{\ast }\right\vert ^{2}\right) \right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert f\left( \frac{m^{2}-m}{2}\left\vert T\right\vert ^{2}\right) \right\vert \right\vert \right\vert. \end{equation*} Також, якщо $f$ є угнутою, зростає на $[0,\infty )$ і $f(0)=0$, то \begin{equation*} \left\vert \left\vert \left\vert f\left( \sum\limits_{i < j}\left\vert T_{ij}\right\vert ^{2}\right) +f\left( \sum\limits_{i < j}\left\vert T_{ji}^{\ast }\right\vert ^{2}\right) \right\vert \right\vert \right\vert \leq (2m^{2}-2m)\left\vert \left\vert \left\vert f\left( \frac{\left\vert T\right\vert ^{2}}{4}\right) \right\vert \right\vert \right\vert. \end{equation*}