Journal of Operator Theory

Volume 63, Issue 1, Winter 2010 pp. 151-157.

Concave functions of positive operators, sums, and congruences

Authors: Jean-Christophe Bourin (1) and Eun-Young Lee (2)
Author institution: (1) Departement de mathematiques, Universite de Franche-Comte, 16 route de Gray, 95030 Besancon, France (2) Department of mathematics, Kyungpook National University, Daegu 702-701, Korea

Summary: Let $A$, $B$, $Z$ be positive semidefinite matrices of same size and suppose $Z$ is expansive, i.e., $Z\geqslant I$. Two remarkable inequalities are $$ \Vert f(A+B)\Vert \leqslant \Vert f(A)+f(B)\Vert \quad {\rm and} \quad \Vert f(ZAZ)\Vert \leqslant \Vert Zf(A)Z\Vert $$ for all non-negative concave function $f$ on $[0,\infty)$ and all symmetric norms $\|\cdot\|$ (in particular for all Schatten $p$-norms). In this paper we survey several related results and we show that these inequalities are two aspects of a unique theorem. For the operator norm, our result also holds for operators on an infinite dimensional Hilbert space.

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