Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

Abstract

Let X and Y be Banach spaces such that each of them is 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 . Let (p, q, r, s) be a quadruple in ${\user2{\mathbb{N}}}$ with p + q  ≥  2 and r + s ≥  2. Suppose that for every pair of Banach spaces X and Y isomorphic to complemented subspaces of each other and satisfying the following Decomposition Scheme

$$ \left\{ \begin{aligned} & X \sim X^{p} \oplus Y^{q}\\ & Y\sim X^{r} \oplus Y^{s}, \end{aligned} \right. $$

we conclude that X^m is isomorphic to Yⁿ for some $m, n \in {\user2{\mathbb{N}}}*$. In this paper, we show that the discriminant $\Delta = (p - 1)(s - 1) - rq$ of this quadruple is different from zero. This result completes the characterization of quadruples in ${\user2{\mathbb{N}}}$ which are nearly Schroeder-Bernstein Quadruples for Banach spaces.