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

 

Subadditivity of eigenvalue sums


Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 134 (2006), 1405-1412
MSC (2000): Primary 47A30, 15A42; Secondary 47A75
Published electronically: October 7, 2005
MathSciNet review: 2199187
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Abstract: Let $ f(t)$ be a nonnegative concave function on $ 0 \leq t <\infty $ with $ f(0)=0$, and let $ X, Y$ be $ n\times n$ matrices. Then it is known that $ \Vert f(\vert X+Y\vert)\Vert_1\leq \Vert f(\vert X\vert)\Vert_1 +\Vert f(\vert Y\vert)\Vert_1$, where $ \Vert \cdot \Vert_1$ is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.


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Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Fukuoka University of Education, Munakata, Fukuoka, 811-4192, Japan
Email: uchiyama@fukuoka-edu.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08116-5
PII: S 0002-9939(05)08116-5
Keywords: Trace, unitarily invariant norm, operator convex function.
Received by editor(s): October 23, 2004
Received by editor(s) in revised form: December 11, 2004
Published electronically: October 7, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society