Relative weak compactness of solid hulls in Banach lattices

Abstract

We show that the solid hull of every relatively weakly compact set in a Banach lattice is again relatively weakly compact if and only if the Banach lattice is an order direct sum of a KB-space and an atomic Banach lattice with an order continuous norm. If we assume order continuity of the norm then this is equivalent to requiring that the image of every relatively weakly compact set under the modulus map is again relatively weakly compact. We also show that amongst Banach lattices with an order continuous norm those that have the property that the lattice operations are weakly sequentially continuous are precisely the atomic ones. The final section of the paper is devoted to applications of our earlier results to questions concerning the factorization of compact and weakly compact operators through reflexive Banach lattices.