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On a Weyl inequality of operators in Banach spaces

Author: Bernd Carl
Journal: Proc. Amer. Math. Soc. 137 (2009), 155-159
MSC (2000): Primary 47B06, 47A75
Published electronically: July 10, 2008
MathSciNet review: 2439436
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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}$,

$\displaystyle \left(\prod\limits_{i=1}^n \vert\lambda_i(T)\vert\right)^{\frac 1... ...{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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Bernd Carl
Affiliation: Mathematisches Institut, FSU Jena, Ernst-Abbe-Platz 1-3, D-07743 Jena, Germany

Keywords: Weyl inequalities, eigenvalue estimates, approximation numbers, $s$-numbers.
Received by editor(s): November 30, 2007
Published electronically: July 10, 2008
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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