Schroeder-Bernstein quintuples for Banach spaces

Let X and Y be two Banach spaces, each isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain necessary and sufficient conditions on the quintuples (p, q, r, s, t) in N for X to be isomorphic to Y whenever [...]. Such quintuples are called Schroeder-Bernstein quintuples for Banach spaces and they yield a unification of the known decomposition methods in Banach spaces involving finite sums of X and Y, similar to Pelczynski's decomposition method. Inspired by this result, we also introduce the notion of Schroeder-Bernstein sextuples for Banach spaces and pose a conjecture which would complete their characterization.