The Schroeder–Bernstein index for Banach spaces

Elói Medina Galego¹

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

Studia Mathematica, Tome 164 (2004) no. 1, pp. 29-38

Résumé

In relation to some Banach spaces recently constructed by W. T. Gowers and B. Maurey, we introduce the notion of Schroeder–Bernstein index ${\rm SBi}(X)$ for every Banach space $X$. This index is related to complemented subspaces of $X$ which contain some complemented copy of $X$. Then we establish the existence of a Banach space $E$ which is not isomorphic to $E^n$ for every $n \in {{\mathbb N}}$, $n \geq 2$, but has a complemented subspace isomorphic to $E^2$. In particular, ${\rm SBi}(E)$ is uncountable. The construction of $E$ follows the approach given in 1996 by W. T. Gowers to obtain the first solution to the Schroeder–Bernstein Problem for Banach spaces.

Détail

DOI : 10.4064/sm164-1-2

Mots-clés : relation banach spaces recently constructed gowers maurey introduce notion schroeder bernstein index sbi every banach space index related complemented subspaces which contain complemented copy establish existence banach space which isomorphic every mathbb geq has complemented subspace isomorphic particular sbi uncountable construction follows approach given gowers obtain first solution schroeder bernstein problem banach spaces