Banach spaces widely complemented in each other

Elói Medina Galego. "Banach spaces widely complemented in each other." Colloquium Mathematicae 133.2 (2013): 283-291. <http://eudml.org/doc/283840>.

@article{ElóiMedinaGalego2013,
abstract = {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 $X^\{p\}$ can be decomposed into a direct sum of $X^\{p-1\}$ 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.},
author = {Elói Medina Galego},
journal = {Colloquium Mathematicae},
keywords = {Schröder-Bernstein problem; complemented subspaces},
language = {eng},
number = {2},
pages = {283-291},
title = {Banach spaces widely complemented in each other},
url = {http://eudml.org/doc/283840},
volume = {133},
year = {2013},
}

TY - JOUR
AU - Elói Medina Galego
TI - Banach spaces widely complemented in each other
JO - Colloquium Mathematicae
PY - 2013
VL - 133
IS - 2
SP - 283
EP - 291
AB - 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 $X^{p}$ can be decomposed into a direct sum of $X^{p-1}$ 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.
LA - eng
KW - Schröder-Bernstein problem; complemented subspaces
UR - http://eudml.org/doc/283840
ER -