A note on extensions of Pełczyński's decomposition method in Banach spaces

Elói Medina Galego. "A note on extensions of Pełczyński's decomposition method in Banach spaces." Studia Mathematica 180.1 (2007): 27-40. <http://eudml.org/doc/286402>.

@article{ElóiMedinaGalego2007,
abstract = {Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ $X^\{u\} ∼ X^\{p\} ⊕ Y^\{q\}$, ⎨ ⎩ $Y^\{v\} ∼ A^\{r\} ⊕ B^\{s\}$. Namely, Ω = (p-u)(s-r-v) - q(r-s) is different from zero and Ω divides p + q - u and v. In other words, we obtain an arithmetic characterization of some extensions of the classical Pełczyński decomposition method in Banach spaces. This result leads naturally to several problems closely related to the Schroeder-Bernstein problem.},
author = {Elói Medina Galego},
journal = {Studia Mathematica},
keywords = {Schroeder-Bernstein problem; Pełczyński’s decomposition method},
language = {eng},
number = {1},
pages = {27-40},
title = {A note on extensions of Pełczyński's decomposition method in Banach spaces},
url = {http://eudml.org/doc/286402},
volume = {180},
year = {2007},
}

TY - JOUR
AU - Elói Medina Galego
TI - A note on extensions of Pełczyński's decomposition method in Banach spaces
JO - Studia Mathematica
PY - 2007
VL - 180
IS - 1
SP - 27
EP - 40
AB - Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ $X^{u} ∼ X^{p} ⊕ Y^{q}$, ⎨ ⎩ $Y^{v} ∼ A^{r} ⊕ B^{s}$. Namely, Ω = (p-u)(s-r-v) - q(r-s) is different from zero and Ω divides p + q - u and v. In other words, we obtain an arithmetic characterization of some extensions of the classical Pełczyński decomposition method in Banach spaces. This result leads naturally to several problems closely related to the Schroeder-Bernstein problem.
LA - eng
KW - Schroeder-Bernstein problem; Pełczyński’s decomposition method
UR - http://eudml.org/doc/286402
ER -