Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that can be decomposed into a direct sum of and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and r ∈ ℕ. Then, we completely determine when X is isomorphic to Y whenever X is (p,q,r) widely complemented in Y and Y is (t,u,v) widely complemented in X. This new notion of complementability leads naturally to an extension of the Square-cube Problem for Banach spaces, the p-q-r Problem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-14,
author = {El\'oi Medina Galego},
title = {Banach spaces widely complemented in each other},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {283-291},
zbl = {1290.46007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-14}
}
Elói Medina Galego. Banach spaces widely complemented in each other. Colloquium Mathematicae, Tome 131 (2013) pp. 283-291. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-14/