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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A matrix subadditivity inequality for symmetric norms


Author: Jean-Christophe Bourin
Journal: Proc. Amer. Math. Soc. 138 (2010), 495-504
MSC (2000): Primary 15A60, 47A30, 47A60
Published electronically: September 11, 2009
MathSciNet review: 2557167
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Abstract: Let $ f(t)$ be a non-negative concave function on $ [0,\infty)$. We prove that

$\displaystyle \Vert f(\vert A+B\vert) \Vert \le \Vert f(\vert A\vert)+f(\vert B\vert) \Vert $

for all normal $ n$-by-$ n$ matrices $ A$, $ B$ and all symmetric norms. This result has several applications. For instance, for a Hermitian $ {\mathbb{A}}=[A_{i, j}]$ partitioned in blocks of the same size,

$\displaystyle \left\Vert f(\vert{\mathbb{A}}\vert) \right\Vert \le \left\Vert \sum f(\vert A_{i, j}\vert) \right\Vert. $

We also prove, in a similar way, that given $ Z$ expansive and $ A$ normal of the same size,

$\displaystyle \Vert f(\vert Z^*AZ\vert) \Vert \le \Vert Z^*f(\vert A\vert)Z \Vert.$


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Additional Information

Jean-Christophe Bourin
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon, France
Email: jcbourin@univ-fcomte.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10103-X
PII: S 0002-9939(09)10103-X
Keywords: Matrix inequalities, symmetric norms, normal operators, concave functions.
Received by editor(s): November 5, 2008
Received by editor(s) in revised form: June 8, 2009
Published electronically: September 11, 2009
Dedicated: Dedicated to Françoise Lust-Piquard, with affection
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.