Schroeder-Bernstein Quintuples for Banach Spaces

Elói Medina Galego. "Schroeder-Bernstein Quintuples for Banach Spaces." Bulletin of the Polish Academy of Sciences. Mathematics 54.2 (2006): 113-124. <http://eudml.org/doc/280362>.

@article{ElóiMedinaGalego2006,
abstract = {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 ⎧$X ~ X^p ⊕ Y^q$, ⎨ ⎩ $Y^t ~ X^r ⊕ Y^s$. 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.},
author = {Elói Medina Galego},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Schroeder-Bernstein problem; square-cube problem},
language = {eng},
number = {2},
pages = {113-124},
title = {Schroeder-Bernstein Quintuples for Banach Spaces},
url = {http://eudml.org/doc/280362},
volume = {54},
year = {2006},
}

TY - JOUR
AU - Elói Medina Galego
TI - Schroeder-Bernstein Quintuples for Banach Spaces
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2006
VL - 54
IS - 2
SP - 113
EP - 124
AB - 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 ⎧$X ~ X^p ⊕ Y^q$, ⎨ ⎩ $Y^t ~ X^r ⊕ Y^s$. 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.
LA - eng
KW - Schroeder-Bernstein problem; square-cube problem
UR - http://eudml.org/doc/280362
ER -