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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Correspondence to Fuad Kittaneh.

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Bourahli, A., Hirzallah, O. & Kittaneh, F. Unitarily invariant norm inequalities for functions of accretive-dissipative \(2\times 2\) block matrices. Positivity (2020). https://doi.org/10.1007/s11117-020-00770-w

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Keywords

  • Accretive-dissipative matrix
  • Concave function
  • Convex function
  • Contraction
  • Schatten p-norm
  • Singular value
  • Unitarily invariant norm

Mathematics Subject Classification

  • 15A18
  • 15A60
  • 47A30
  • 47B15