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

2019/09/20 by Botelho, Geraldo, Luiz, José Lucas P.
#46A04 #46A40 #46B42 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1909.09514

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.

Related

• Polynomial Schur and Polynomial Dunford-Pettis Properties
• Disjoint Dunford-Pettis-type properties in Banach lattices
• Reciprocal Dunford--Pettis sets and V^*-sets in Banach lattices
• On a Schur-like property for spaces of measures
• The positive polynomial Schur property in Banach lattices