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Norm inequalities involving accretive-dissipative $$2\times 2$$ block matrices. (English) Zbl 1398.15020
Summary: Let $$T_{11}, T_{12}, T_{21}$$, and $$T_{22}$$ be $$n \times n$$ complex matrices, and let $$H = \left (\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix} \right)$$ be accretive-dissipative. It is shown that if $$f$$ is an increasing convex function on $$[0, \infty)$$ such that $$f(0) = 0$$, then $$||| f(| T_{12} |^2) + f(| T_{21}^\ast |^2) ||| \leq ||| f(| T |^2) |||$$ for every unitarily invariant norm $$||| \cdot |||$$. Moreover, if $$f$$ is an increasing concave function on $$[0, \infty)$$ such that $$f(0) = 0$$, then $$||| f(| T_{12} |^2) + f(| T_{21}^\ast |^2) ||| \leq 4 ||| f(\frac{| T |^2}{4}) |||$$ for every unitarily invariant norm $$||| \cdot |||$$. Among other inequalities for the Schatten $$p$$-norms, it is shown that$$\| T_{12} \|_p^p + \| T_{21} \|_p^p \leq 2^{p - 1} \| T_{11} \|_p^{p / 2} \| T_{22} \|_p^{p / 2}$$ for $$p \geq 2$$.

##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A18 Eigenvalues, singular values, and eigenvectors 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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