Journal of Operator Theory
Volume 63, Issue 1, Winter 2010 pp. 151-157.
Concave functions of positive operators, sums, and congruencesAuthors: 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⩾. 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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