A note on extensions of Pełczyński's decomposition method in Banach spaces

Elói Medina Galego¹

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

Studia Mathematica, Tome 180 (2007) no. 1, pp. 27-40

Résumé

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: $$\left \{\eqalign{ ^u \sim X^p \oplus Y^q, \cr ^v \sim A^r \oplus B^s.\cr } \right.$$ 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.

Détail

DOI : 10.4064/sm180-1-3

Mots-clés : banach spaces isomorphic oplus isomorphic oplus gowers solved schroeder bernstein problem banach spaces showing necessarily isomorphic present paper necessary sufficient condition sextuples nbsp mathbb geq geq mathbb * isomorphic whenever these spaces satisfy following decomposition scheme eqalign sim oplus sim oplus right namely mit omega p u s r v q r s different zero mit omega divides q u other words obtain arithmetic characterization extensions classical czy ski decomposition method banach spaces result leads naturally several problems closely related schroeder bernstein problem