# Optimal Weyl-type Inequalities for Operators in Banach Spaces

Optimal Weyl-type Inequalities for Operators in Banach Spaces Let (s n ) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality [InlineMediaObject not available: see fulltext.]between the eigenvalues and s-numbers of a compact operator T in a Banach space. Furthermore, the constant (k + 1)1/2 is optimal for n = k + 1 and k = 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand we prove that the Weyl numbers form a minimal multiplicative s-number sequence and by a well-known inequality between eigenvalues and Weyl numbers due to A. Pietsch they are very good quantities for investigating the optimal asymptotic behavior of eigenvalues. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Optimal Weyl-type Inequalities for Operators in Banach Spaces

, Volume 11 (1) – Oct 13, 2006
15 pages

/lp/springer_journal/optimal-weyl-type-inequalities-for-operators-in-banach-spaces-WtGomHdSsX
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-006-1088-0
Publisher site
See Article on Publisher Site

### Abstract

Let (s n ) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality [InlineMediaObject not available: see fulltext.]between the eigenvalues and s-numbers of a compact operator T in a Banach space. Furthermore, the constant (k + 1)1/2 is optimal for n = k + 1 and k = 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand we prove that the Weyl numbers form a minimal multiplicative s-number sequence and by a well-known inequality between eigenvalues and Weyl numbers due to A. Pietsch they are very good quantities for investigating the optimal asymptotic behavior of eigenvalues.

### Journal

PositivitySpringer Journals

Published: Oct 13, 2006

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