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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Operators on $C(K)$ spaces preserving copies of Schreier spaces


Author: Ioannis Gasparis
Journal: Trans. Amer. Math. Soc. 357 (2005), 1-30
MSC (2000): Primary 46B03; Secondary 06A07, 03E02
Published electronically: August 19, 2004
MathSciNet review: 2098084
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Abstract: It is proved that an operator \(T \colon C(K) \to X\), \(K\)compact metrizable, \(X\) a separable Banach space, for which the \(\epsilon\)-Szlenk index of \(T^*(B_{X^*})\) is greater than or equal to \(\omega^\xi\), \(\xi < \omega_1\), is an isomorphism on a subspace of \(C(K)\) isomorphic to \(X_\xi\), the Schreier space of order \(\xi\). As a corollary, one obtains that a complemented subspace of \(C(K)\) with Szlenk index equal to \(\omega^{\xi + 1}\) contains a subspace isomorphic to \(X_\xi\).


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Additional Information

Ioannis Gasparis
Affiliation: Department of Mathematics, University of Crete, Knossou Avenue, P.O. Box 2208, Heracleion 71409, Greece
Address at time of publication: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
Email: ioagaspa@math.uoc.gr, iogaspa@auth.gr

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03688-8
PII: S 0002-9947(04)03688-8
Keywords: $C(K)$ space, Szlenk index, projection, Schreier sets
Received by editor(s): February 4, 2003
Published electronically: August 19, 2004
Article copyright: © Copyright 2004 American Mathematical Society