Some Schroeder–Bernstein type theorems for Banach spaces

We first introduce the notion of (p, q, r)-complemented subspaces in Banach spaces, where p, q, r ∈ N. Then, given a couple of triples {(p, q, r), (s, t,u)} in N and putting Λ = (q + r − p)(t + u − s) − ru, we prove partially the following conjecture: For every pair of Banach spaces X and Y such that X is (p, q, r)-complemented in Y and Y is (s, t,u)-complemented in X, we have that X is isomorphic Y if and only if one of the following conditions holds: (a) Λ = 0, Λ divides p −q and s − t , p = 1 or q = 1 or s = 1 or t = 1. (b) p = q = s = t = 1 and gcd(r, u) = 1. The case {(2, 1, 1), (2, 1, 1)} is the well-known Pełczy´nski’s decomposition method. Our result leads naturally to some generalizations of the Schroeder–Bernstein problem for Banach spaces solved by W.T. Gowers in 1996. © 2007 Elsevier Inc. All rights reserved