# Unitarily invariant norm inequalities for functions of accretive-dissipative $$2\times 2$$ block matrices

## Abstract

Let $$T_{11},T_{12},T_{21},$$ and $$T_{22}$$ be $$n\times n$$ complex matrices, $$\ T=\left( \begin{array}{cc} T_{11} &{} T_{12} \\ T_{21} &{} T_{22} \end{array} \right)$$ be accretive-dissipative, $$\gamma \in (0,1]$$, $$r\ge 2,\$$and let $$\alpha ,\beta \in [0,1]$$ such that $$\alpha +\beta =1$$. If f is an increasing convex function on $$[0,\infty )$$ such that $$f(0)=0$$, then

\begin{aligned} \left| \left| \left| f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{r}\right) +f\left( 2^{r}\alpha ^{r/2}\beta ^{r/2}\left| T_{21}^{*}\right| ^{r}\right) \right| \right| \right| \le \gamma \left| \left| \left| f\left( \frac{\left| T\right| ^{r}}{\gamma ^{r/2}}\right) \right| \right| \right| \end{aligned}

for every unitarily invariant norm. In addition, if T is contraction and f is submultiplicative, then

\begin{aligned}&\left| \left| \left| \left( f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{2}\right) +f\left( 4\alpha \beta \left| T_{21}^{*}\right| ^{2}\right) \right) \right| \right| \right| \\&\quad \le f^{2}\left( \sqrt{2}\right) \left| \left| \left| f^{r}\left( \left| T_{11}\right| ^{1/2}\right) \right| \right| \right| ^{1/r}\left| \left| \left| f^{s}\left( \left| T_{22}\right| ^{1/2}\right) \right| \right| \right| ^{1/s} \end{aligned}

for every unitarily invariant norm and for every positive real numbers rs with $$\frac{1}{r}+\frac{1}{s}=1$$. Related inequalities for concave functions are also given.

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## Acknowledgements

The authors are grateful to the referee for his comments and suggestions.

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