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A matrix subadditivity inequality for \(f(A + B)\) and \(f(A) + f(B)\). (English) Zbl 1123.15013

Authors’ abstract: T. Ando and X. Zhan [Math. Ann. 315, No. 4, 771–780 (1999; Zbl 0941.47004)] proved a subadditivity inequality for operator concave functions. We extend it to all concave functions: Given positive semidefinile matrices \(A,B\) and a non-negative concave function \(f\) on \([0,\infty)\), \[ \|f(A+B)\|\leq\|f(A)+ f(B)\| \] for all symmetric norms (in particular for all Schatten \(p\)-norms). The case \(f(t)=\sqrt t\) is connected to some block-matrix inequalities, for instance the operator norm inequality \[ \left\|\left( \begin{matrix} A & X^*\\ X& B\end{matrix}\right)\right\|_\infty \leq\max\{\| |A|+ |X|\|_\infty;\||B|+|X^*|\|_\infty\} \] for any partitioned Hermitian matrix.

MSC:

15A45 Miscellaneous inequalities involving matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A30 Norms (inequalities, more than one norm, etc.) of linear operators

Citations:

Zbl 0941.47004
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References:

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