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Metric Entropy Conditions for Kernels, Schatten Classes and Eigenvalue Problems

  • Bernd Carl
Chapter

Abstract

In this paper we investigate the problem how the metric entropy of the image Im(K) of a bounded ‘abstract kernel’ K: XE′ mapping an arbitrary set X into the dual E′ of a Banach space E reflects the rate of decay of approximation quantities of the induced operator
$$ ({T_K}x)(s): = < x,K(s) > {\text{for}}x \in E{\text{ and }}s \in X, $$
considered from E into l (X). In the case of Hilbert spaces, we give sufficient and optimal conditions for the metric entropy of Im(K) which guarantee that the induced integral operator T K : L 2 (Y, v) → L 2(X, μ), where (X, μ), (Y, v) are finite measure spaces, belongs to the Schatten classes S q,t . In order to illustrate the usefulness of our results we apply them to eigenvalue problems.

Keywords

Banach Space Integral Operator Compact Operator Entropy Condition Entropy Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    B. Carl and I. Stephani, “Entropy, Compactness and the Approximation of Operators”, Cambridge University Press, 1990.zbMATHGoogle Scholar
  2. [2]
    J. M. Gonzales-Barrios and R. M. Dudley, “Metric entropy conditions for an operator, to be of trace class”, Proc. Amer. Math. Soc., 118, 1993, 175–180.MathSciNetGoogle Scholar
  3. [3]
    A. Pietsch, Eggenvalues and s-Numbers, Leipzig: Geest and Portig K.-G., 1987.Google Scholar
  4. [4]
    C. Richter, “Entropy, approximation quantities and the asymptotics of the modulus of continuity”, Math. Nachr.,(to appear).Google Scholar
  5. [5]
    C. Richter and I. Stephani, “Entropy and the approximation of bounded functions and operators”, Arch. Math., 67, 1996, 478–492.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Bernd Carl
    • 1
  1. 1.Fakultät für Mathematik und InformatikUniversität JenaJenaGermany

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