Some Schroeder–Bernstein type theorems for Banach spaces

Abstract

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 ) − r u , 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.