On the Schur, positive Schur and weak Dunford-Pettis properties in Fr\'echet lattices

TL;DR

This paper investigates the relationships between various convergence and positivity properties in Fréchet lattices and their ideals, establishing conditions under which these properties are preserved or transferred.

Contribution

It generalizes known properties from Banach lattices to Fréchet lattices, providing new results on the transfer of the positive Schur, Schur, and weak Dunford-Pettis properties.

Findings

If two of E, I, E/I have the positive Schur property, the third also has it.

If I and E/I have the dual positive Schur property, E inherits this property.

If I has the weak Dunford-Pettis property and E/I has the positive Schur property, then E also has the weak Dunford-Pettis property.

Abstract

We prove some general results on sequential convergence in Fr\'echet 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.