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 Pełczyński'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.