A note on weak almost limited operators


Machrafi N., El Fahri K., Moussa M., ALTIN B.

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, vol.48, no.3, pp.759-770, 2019 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48 Issue: 3
  • Publication Date: 2019
  • Doi Number: 10.15672/hjms.2018.550
  • Journal Name: HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
  • Journal Indexes: Science Citation Index Expanded, Scopus, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.759-770
  • Keywords: weak almost limited operator, weak* Dunford-Pettis operator, weak Dunford-Pettis* property, Banach lattice, L SETS

Abstract

Let us recall that an operator T : E -> F, between two Banach lattices, is said to be weak* Dunford-Pettis (resp. weak almost limited) if f(n)(Tx(n)) -> 0 whenever (x(n)) converges weakly to 0 in E and (f(n)) converges weak* to 0 in F' (resp. fn (Tx(n)) -> 0 for all weakly null sequences (x(n)) subset of E and all weak* null sequences (f(n)) subset of F' with pairwise disjoint terms). In this note, we state some sufficient conditions for an operator R : G -> E(resp. S : F -> G), between Banach lattices, under which the product TR (resp. ST) is weak* Dunford-Pettis whenever T : E -> F is an order bounded weak almost limited operator. As a consequence, we establish the coincidence of the above two classes of operators on order bounded operators, under a suitable lattice operations' sequential continuity of the spaces (resp. their duals) between which the operators are defined. We also look at the order structure of the vector space of weak almost limited operators between Banach lattices.