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

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## Abstract.

Suppose that then the classical Pełczyński’s decomposition method for Banach spaces shows 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,$$ \left\{ \begin{array}{l} X \sim X^2 \oplus A\\ Y \sim Y^2 \oplus B, \end{array}\right.$$

*X*is isomorphic to*Y*whenever we can assume that*A*=*B*= {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## Preview

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## Copyright information

© Birkhaeuser 2008