Cantor-Bernstein theorems for Orlicz sequence spaces

Carlos E. Finol, Marcos J. González, and Marek Wójtowicz. "Cantor-Bernstein theorems for Orlicz sequence spaces." Banach Center Publications 102.1 (2014): 71-88. <http://eudml.org/doc/286637>.

@article{CarlosE2014,
abstract = {For two Banach spaces X and Y, we write $dim_\{ℓ\}(X) = dim_\{ℓ\}(Y)$ if X embeds into Y and vice versa; then we say that X and Y have the same linear dimension. In this paper, we consider classes of Banach spaces with symmetric bases. We say that such a class ℱ has the Cantor-Bernstein property if for every X,Y ∈ ℱ the condition $dim_\{ℓ\}(X) = dim_\{ℓ\}(Y)$ implies the respective bases (of X and Y) are equivalent, and hence the spaces X and Y are isomorphic. We prove (Theorems 3.1, 3.3, 3.5) that the class of Orlicz sequence spaces generated by regularly varying Orlicz functions is of this type. This complements some results in this direction obtained earlier by S. Banach (Proposition 1.1), L. Drewnowski (Proposition 1.2), and M. J. Gonzalez, B. Sari and M. Wójtowicz (Theorem 1.4). Our theorems apply to large families of concrete Orlicz spaces.},
author = {Carlos E. Finol, Marcos J. González, Marek Wójtowicz},
journal = {Banach Center Publications},
keywords = {linear dimension; symmetric bases; Orlicz sequence space},
language = {eng},
number = {1},
pages = {71-88},
title = {Cantor-Bernstein theorems for Orlicz sequence spaces},
url = {http://eudml.org/doc/286637},
volume = {102},
year = {2014},
}

TY - JOUR
AU - Carlos E. Finol
AU - Marcos J. González
AU - Marek Wójtowicz
TI - Cantor-Bernstein theorems for Orlicz sequence spaces
JO - Banach Center Publications
PY - 2014
VL - 102
IS - 1
SP - 71
EP - 88
AB - For two Banach spaces X and Y, we write $dim_{ℓ}(X) = dim_{ℓ}(Y)$ if X embeds into Y and vice versa; then we say that X and Y have the same linear dimension. In this paper, we consider classes of Banach spaces with symmetric bases. We say that such a class ℱ has the Cantor-Bernstein property if for every X,Y ∈ ℱ the condition $dim_{ℓ}(X) = dim_{ℓ}(Y)$ implies the respective bases (of X and Y) are equivalent, and hence the spaces X and Y are isomorphic. We prove (Theorems 3.1, 3.3, 3.5) that the class of Orlicz sequence spaces generated by regularly varying Orlicz functions is of this type. This complements some results in this direction obtained earlier by S. Banach (Proposition 1.1), L. Drewnowski (Proposition 1.2), and M. J. Gonzalez, B. Sari and M. Wójtowicz (Theorem 1.4). Our theorems apply to large families of concrete Orlicz spaces.
LA - eng
KW - linear dimension; symmetric bases; Orlicz sequence space
UR - http://eudml.org/doc/286637
ER -