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).References
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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