Abstract
Let (s n ) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality 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.
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Research of the second author was supported by the DFG Emmy-Noether grant Hi 584/2-3.
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Carl, B., Hinrichs, A. Optimal Weyl-type Inequalities for Operators in Banach Spaces. Positivity 11, 41–55 (2007). https://doi.org/10.1007/s11117-006-1088-0
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DOI: https://doi.org/10.1007/s11117-006-1088-0