In relation to some Banach spaces recently constructed by W. T. Gowers and B. Maurey, we introduce the notion of Schroeder-Bernstein index 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 Eⁿ for every n ∈ ℕ, n ≥ 2, but has a complemented subspace isomorphic to E². In particular, 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-1-2,
author = {El\'oi Medina Galego},
title = {The Schroeder-Bernstein index for Banach spaces},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {29-38},
zbl = {1081.46006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-1-2}
}
Elói Medina Galego. The Schroeder-Bernstein index for Banach spaces. Studia Mathematica, Tome 162 (2004) pp. 29-38. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-1-2/