Embeddings of rearrangement invariant spaces that are not strictly singular

We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L1([0, 1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space LΦ with Φ(x) = exp(x 2) − 1. In this paper we ask the following question. Given a rearrangement invariant space E on [0, 1], when is the natural embedding E ⊂ L 1 ([0, 1]) strictly singular. (We refer the reader to [4] for the definition and properties of rearrangement invariant spaces.) We define a linear map between two normed spaces to be strictly singular if there does not exist an infinite dimensional subspace of the domain upon which the operator is an isomorphism. This question is a natural extension of similar work by del Amo, Hernández, Sánchez and Semenov [1], when they considered the problem of which embeddings between rearrangement invariant spaces are not disjointly strictly singular. A positive linear operator between two Banach lattices is disjointly strictly singular if there exists an infinite sequence of non-zero disjoint elements in the domain such that the operator is an isomorphism on the span of this sequence. This work [1] contains a number of very sharp results, giving some very clear criteria. However the question concerning when such maps are strictly singular seems to be more difficult. For this reason, we will restrict ourselves to considering the case when the range is L 1 ([0, 1]). Even then, we do not have complete answers, and in this paper, we leave as many questions unanswered as we answer. The " other end " was investigated by Novikov [7], who showed that the natural embedding L ∞ ([0, 1]) ⊂ E is strictly singular unless E is