This is an announcement for the paper "Estimates for covering numbers in Schauder's theorem about adjoints of compact operators" by Michael Cwikel and Eliahu Levy.
Abstract: Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y' --> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y' respectively. We obtain apparently new estimates for the covering numbers of the set S(B(Y')). These are expressed in terms of the covering numbers of T(B(X)), or, more generally, in terms of the covering numbers of a "significant" subset of T(B(X)). The latter more general estimates are best possible. These estimates follow from our new quantitative version of an abstract compactness result which generalizes classical theorems of Arzela-Ascoli and of Schauder. Analogous estimates also hold for the covering numbers of T(B(X)), in terms of the covering numbers of S(B(Y')) or in terms of a suitable "significant" subset of S(B(Y')).
Archive classification: math.FA
Mathematics Subject Classification: Primary 46B06. Secondary 46B10, 46B50, 05B40, 52C17, 52C15.
Remarks: 14 pages. At any given time our most recent version of this paper will be either at http://www.math.technion.ac.il/~mcwikel/compact/QuantitativeSchauder.pdf or http://arxiv.org/abs/0810.4240
The source file(s), 8QuantitativeSchauder.tex: 51761 bytes, is(are) stored in gzipped form as 0810.4240.gz with size 15kb. The corresponding postcript file has gzipped size 105kb.
Submitted from: mcwikel@math.technion.ac.il
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http://front.math.ucdavis.edu/0810.4240
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http://arXiv.org/abs/0810.4240
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