On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

Abstract

We prove some general results on sequential convergence in Fréchet lattices that yield, as particular instances, the following results regarding a closed ideal \(I\) of a Banach lattice \(E\): (i) If two of the lattices \(E\), \(I\) and \(E/I\) have the positive Schur property (the Schur property, respectively) then the third lattice has the positive Schur property (the Schur property, respectively) as well; (ii) If \(I\) and \(E/I\) have the dual positive Schur property, then \(E\) also has this property; (iii) If \(I\) has the weak Dunford-Pettis property and \(E/I\) has the positive Schur property, then \(E\) has the weak Dunford-Pettis property. Examples and applications are provided.