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 X2, we introduce the concept of P-complemented subspaces in Banach spaces. In this way, the well-known Pełczyński'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.