Banach spaces widely complemented in each other

Abstract

Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X 2 , then X is isomorphic to Y whenever there exists p ∈ N such that X p can be decomposed into a direct sum of X p−1 and Y . Motivated by this fact, we introduce the concept of (p, q, r) widely complemented subspaces in Banach spaces, where p, q and r ∈ N. Then, we completely determine when X is isomorphic to Y whenever X is (p, q, r) widely complemented in Y and Y is (t, u, v) widely complemented in X. This new notion of complementability leads naturally to an extension of the Square-cube Problem for Banach spaces, the p-q-r Problem.