A family of Schroeder-Bernstein type theorems for Banach spaces

Motivated by a characterization of the complemented subspaces in Banach spaces X isomorphic to their squares X-2, we introduce the concept of P-complemented subspaces in Banach spaces. In this way, the well-known Pelczynski`s decomposition method can be seen as a Schroeder-Bernstein type theorem. Then, we give a complete description of the Schroeder-Bernstein type theorems for this new notion of complementability. By contrast, some very elementary questions on P-complementability are refinements of the Square-Cube Problem closely connected with some Banach spaces introduced by W.T. Gowers and B. Maurey in 1997. (C) 2007 Elsevier Inc. All rights reserved.