Cantor–Bernstein theorems for Orlicz sequence spaces

Abstract

For two Banach spaces X and Y , we write dim (X) = dim (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 (X) = dim (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. Introduction.In what follows, we use the notation from the abstract.The study of the comparison of the linear dimension between Banach spaces goes back to S. Banach himself (see [B, Chap. XII, p. 193]). For example, in [B, Chap. XII,

The structure of symmetric basic sequences with applications to a class of Orlicz sequence spaces

“…The sets C M , where M is either regularly varying or rapidly varying, have been computed in[8, Lemma 4.3].Proposition 2.13. Let M be a non-degenerate Orlicz function such that the limit f (t) = lim s→0 + M (st) M (s) exists, for every t ≥ 0.…”