Quantitative positive Schur property in Banach lattices

Abstract

We introduce and investigate quantitative versions of the positive Schur property. We show, in particular, that each L 1 ( μ ) L_{1}(\mu ) -space( μ \mu σ \sigma -finite) enjoys the 1 1 -positive Schur property in the strongest possible way. By introducing a quantity measuring L L -weak non-compactness of sets in Banach lattices, we prove that the positive Schur property is always quantitative. We also show that this quantity is equal to several natural measures of weak non-compactness in abstract L L -spaces.