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

摘要

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, { l X ~ X2 ÅAY ~ Y2 ÅB, left{ begin{array}{l} X sim X^2 oplus A Y sim Y^2 oplus B, end{array}right.