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 r, s with $\frac{1}{r}+\frac{1}{s}=1$. Related inequalities for concave functions are also given.