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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a Weyl inequality of operators in Banach spaces
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by Bernd Carl PDF
Proc. Amer. Math. Soc. 137 (2009), 155-159 Request permission

Abstract:

Let $s=(s_n)$ be an injective and surjective $s$-number sequence in the sense of Pietsch. We show for a Riesz-operator $T:X\to X$ acting on a (complex) Banach space the following Weyl inequality between geometric means of eigenvalues and $s$-numbers: For any $0<\delta \le 1$ and all $n\in \mathbb {N}$, \[ \left (\prod \limits _{i=1}^n |\lambda _i(T)|\right )^{\frac 1n} \le c_0\left (1+\frac 1{\delta } \ln \left (\frac 1 {\delta }\right )\right ) \left (\prod \limits _{i=1}^{\left [\frac n {1+\delta }\right ]} s_i(T)\right )^{\frac 1{\left [\frac n {1+\delta }\right ]}}~, \] where $c_0\ge 1$ is an absolute constant. The proof rests on an elementary mixing multiplicativity of an arbitrary $s$-number sequence and a striking result of G. Pisier. The inequality is a contribution to the problem of estimating eigenvalues by $s$-numbers first started in a strong sense by H. König (1986, 2001).
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Additional Information
  • Bernd Carl
  • Affiliation: Mathematisches Institut, FSU Jena, Ernst-Abbe-Platz 1-3, D-07743 Jena, Germany
  • Email: carl@minet.uni-jena.de
  • Received by editor(s): November 30, 2007
  • Published electronically: July 10, 2008
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 155-159
  • MSC (2000): Primary 47B06, 47A75
  • DOI: https://doi.org/10.1090/S0002-9939-08-09532-4
  • MathSciNet review: 2439436