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AN ARITHMETICAL CHARACTERIZATION OF DECOMPOSITION METHODS IN BANACH SPACES VIA SUPPLEMENTED SUBSPACES

Published online by Cambridge University Press:  29 November 2005

ELÓI MEDINA GALEGO
ELÓI MEDINA GALEGO
Affiliation:
Department of Mathematics - IME, University of São Paulo, São Paulo 05315-970, Brazil e-mail: eloi@ime.usp.br
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Abstract

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Let X and Y be Banach spaces such that each one is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we give suitable conditions on X, Y, the supplemented subspaces of Y in X and of X in Y to yield that X is isomorphic to Y. In other words, we obtain generalizations of Pełczyński's decomposition method via supplemented subspaces. In order to determine all the possible generalizations, we introduce the notion of Mixed Schroeder-Bernstein Quadruples for Banach spaces. Then, we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.

Keywords

46B0346B20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue 3 , September 2005 , pp. 489 - 500
Copyright
2005 Glasgow Mathematical Journal Trust