A note on weak almost limited operators


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

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, cilt.48, sa.3, ss.759-770, 2019 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 48 Konu: 3
  • Basım Tarihi: 2019
  • Doi Numarası: 10.15672/hjms.2018.550
  • Dergi Adı: HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
  • Sayfa Sayıları: ss.759-770

Özet

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.