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-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 48 Sayı: 3
  • Basım Tarihi: 2019
  • Doi Numarası: 10.15672/hjms.2018.550
  • Dergi Adı: HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
  • Sayfa Sayıları: ss.759-770
  • Anahtar Kelimeler: weak almost limited operator, weak* Dunford-Pettis operator, weak Dunford-Pettis* property, Banach lattice, L SETS
  • Gazi Üniversitesi Adresli: Evet

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.