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 ℕ for X to be isomorphic to Y whenever ⎧, ⎨ ⎩ . 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 Pełczyński’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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-2-3,
author = {El\'oi Medina Galego},
title = {Schroeder-Bernstein Quintuples for Banach Spaces},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {54},
year = {2006},
pages = {113-124},
zbl = {1109.46011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-2-3}
}
Elói Medina Galego. Schroeder-Bernstein Quintuples for Banach Spaces. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) pp. 113-124. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-2-3/