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Abstract

Let $X,Y,A$ and $B$ be Banach spaces such that $X$ is isomorphic to $Y\oplus A$ and $Y$ is isomorphic to $X\oplus B$. In 1996, W. T. Gowers solved the Schroeder–Bernstein problem for Banach spaces by showing that $X$ is not necessarily isomorphic to $Y$. In the present paper, we give a necessary and sufficient condition on sextuples $(p, q, r, s, u, v)$ in ${\mathbb N}$ with $p+q \geq 2$, $r+s \geq 1$ and $u, v \in {\mathbb N}^*$ for $X$ to be isomorphic to $Y$ whenever these spaces satisfy the following decomposition scheme: $$\left \{\eqalign{ &X^u \sim X^p \oplus Y^q, \cr &Y^v \sim A^r \oplus B^s.\cr } \right.$$ Namely, ${\mit\Omega}=(p-u)(s-r-v)-q(r-s)$ is different from zero and ${\mit\Omega}$ divides $p+q-u$ and $v$. In other words, we obtain an arithmetic characterization of some extensions of the classical Pełczyński decomposition method in Banach spaces. This result leads naturally to several problems closely related to the Schroeder–Bernstein problem.