Cantor–Bernstein theorems for Orlicz sequence spaces
Volume 102 / 2014
Abstract
For two Banach spaces $X$ and $Y$, we write $\dim _\ell(X)= \dim _\ell(Y)$ if $X$ embeds into $Y$ and vice versa; then we say that \emph{$X$ 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 $\mathcal F$ has the Cantor–Bernstein property if for every $X,Y\in {\cal F}$ the condition $\dim _\ell(X)= \dim _\ell(Y)$ 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.
