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

Abstract

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 Pelczy ´ nski'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.