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*}
Key words: Accretive-dissipative matrix; convex function; concave function; inequality; singular value; unitarily invariant norm.
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