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Optimal Weyl inequality in Banach spaces

Author(s): Aicke Hinrichs
Journal: Proc. Amer. Math. Soc. 134 (2006), 731-735.
MSC (2000): Primary 47B10, 43A25
Posted: July 18, 2005
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Abstract: A well-known multiplicative Weyl inequality states that the sequence of eigenvalues $(\lambda_k(T))$ and the sequence of approximation numbers $(a_k(T))$ of any compact operator $T$ in a Banach space satisfy

\begin{displaymath}\prod_{k=1}^n \vert\lambda_k(T)\vert \le n^{n/2} \prod_{k=1}^n a_k(T)\end{displaymath}

for all $n$. We prove here that the constant $n^{n/2}$ is optimal, which solves a longstanding problem.

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Additional Information:

Aicke Hinrichs
Affiliation: Mathematisches Institut, FSU Jena, Ernst-Abbe-Platz 1-3, D-07743 Jena, Germany
Email: hinrichs@minet.uni-jena.de

DOI: 10.1090/S0002-9939-05-08019-6
PII: S 0002-9939(05)08019-6
Keywords: Weyl inequality, eigenvalue estimates, approximation numbers, $s$-numbers.
Received by editor(s): October 6, 2004
Posted: July 18, 2005
Additional Notes: The research of the author was supported by the DFG Emmy-Noether grant Hi 584/2-3.
Dedicated: Dedicated to Professor Albrecht Pietsch on the occasion of his {70}th birthday
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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