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

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 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.