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Archiv der Mathematik

, Volume 90, Issue 6, pp 530–536 | Cite as

Generalizations of Pełczyński’s decomposition method for Banach spaces containing a complemented copy of their squares

  • Elói Medina Galego
Article
  • 69 Downloads

Abstract.

Suppose that X and Y are Banach spaces isomorphic to complemented subspaces of each other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. However, if X 2 is complemented in X with supplement A and Y 2 is complemented in Y with supplement B, that is,
$$ \left\{ \begin{array}{l} X \sim X^2 \oplus A\\ Y \sim Y^2 \oplus B, \end{array}\right.$$
then the classical Pełczyński’s decomposition method for Banach spaces shows that X is isomorphic to Y whenever we can assume that AB = {0}. But unfortunately, this is not always possible. In this paper, we show that it is possible to find all finite relations of isomorphism between A and B which guarantee that X is isomorphic to Y. In order to do this, we say that a quadruple (p, q, r, s) in \({\mathbb{N}}\) is a P-Quadruple for Banach spaces if X is isomorphic to Y whenever the supplements A and B satisfy \(A^p {\oplus} B^q {\sim} A^r {\oplus} B^s\). Then we prove that (p, q, r, s) is a P-Quadruple for Banach spaces if and only if p − r = s − q = ±1.

Mathematics Subject Classification (2000).

Primary 46B03, 46B20 

Keywords.

Pełczyński’s decomposition method Schroeder-Bernstein problem 

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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Department of Mathematics - IMEUniversity of São PauloSão PauloBrazil

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