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Abstract

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 ${\mathbb N}$ for $X$ to be isomorphic to $Y$ whenever $$ \cases{ X \sim X^p \oplus Y^q, \cr Y^t \sim X^r \oplus Y^{s}. }$$ 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 Pe/lczyński'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.