Schroeder–Bernstein Quintuples for Banach Spaces

Elói Medina Galego¹

¹ Department of Mathematics – IME University of São Paulo São Paulo 05315-970, Brazil

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) no. 2, pp. 113-124

Résumé

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.

Détail

DOI : 10.4064/ba54-2-3

Keywords: banach spaces each isomorphic complemented subspace other gowers solved schroeder bernstein problem banach spaces showing necessarily isomorphic paper obtain necessary sufficient conditions quintuples mathbb isomorphic whenever cases sim oplus sim oplus quintuples called schroeder bernstein quintuples banach spaces yield unification known decomposition methods banach spaces involving finite sums similar lczy skis decomposition method inspired result introduce notion schroeder bernstein sextuples banach spaces pose conjecture which would complete their characterization