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.Definition 1.1. A property P of Banach lattices is a 3-lattice property if the following holds: given a closed ideal I of the Banach lattice E, if two of the lattices E, I and E/I have P, then the third one has P as well.A Banach lattice has the positive Schur property if weakly null sequences formed by positive vectors are norm null. A lot of research has been done on this property, for some recent contributions see, e.g., [6,7,13,18,24,25,26,28]. Among other results, in this paper we prove that, contrary to the case of the Schur property for Banach spaces, the Schur and the positive Schur properties are 3-lattice properties. These results will appear as applications of general results on the Schur and the positive Schur properties in Fréchet lattices.