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.


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


Zbl 0941.47004
Full Text: DOI arXiv


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