Weakly compact sets and weakly compact pointwise multipliers in Banach function lattices

TL;DR

This paper characterizes Banach function lattices where weakly compact sets are equi-integrable, showing they are exactly the 1-disjointly homogeneous spaces, and applies these results to weakly compact pointwise multipliers.

Contribution

It establishes a precise equivalence between weakly compact sets and equi-integrability in Banach lattices, introduces new examples, and characterizes weakly compact multipliers.

Findings

Spaces satisfying Dunford-Pettis criterion are 1-disjointly homogeneous.

Dunford-Pettis criterion is equivalent to de la Vallée Poussin criterion in rearrangement invariant spaces.

Characterization of weakly compact pointwise multipliers between Banach function lattices.

Abstract

We prove that the class of Banach function lattices in which all relatively weakly compact sets are equi-integrable sets (i.e. spaces satisfying the Dunford-Pettis criterion) coincides with the class of 1-disjointly homogeneous Banach lattices. A new examples of such spaces are provided. Furthermore, it is shown that Dunford-Pettis criterion is equivalent to de la Vallee Poussin criterion in all rearrangement invariant spaces on the interval. Finally, the results are applied to characterize weakly compact pointwise multipliers between Banach function lattices.