The Schroeder–Bernstein index for Banach spaces
Volume 164 / 2004
Studia Mathematica 164 (2004), 29-38
MSC: Primary 46B03, 46B20.
DOI: 10.4064/sm164-1-2
Abstract
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.
