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Optimal Weyl inequality in Banach spaces

Author: Aicke Hinrichs
Journal: Proc. Amer. Math. Soc. 134 (2006), 731-735
MSC (2000): Primary 47B10, 43A25
Published electronically: July 18, 2005
MathSciNet review: 2180891
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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

Keywords: Weyl inequality, eigenvalue estimates, approximation numbers, $s$-numbers.
Received by editor(s): October 6, 2004
Published electronically: 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 70th birthday
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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