Metric Entropy Conditions for Kernels, Schatten Classes and Eigenvalue Problems

  • Bernd Carl


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.


Banach Space Integral Operator Compact Operator Entropy Condition Entropy Number 
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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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