Domination by Positive Weak Dunford–pettis Operators on Banach Lattices

Abstract

Recently, H'michane et al. ['On the class of limited operators', Acta Math. Sci. (submitted)] introduced the class of weak * Dunford-Pettis operators on Banach spaces, that is, operators which send weakly compact sets onto limited sets. In this paper, the domination problem for weak * Dunford-Pettis operators is considered. Let S , T : E → F be two positive operators between Banach lattices E and F such that 0 ≤ S ≤ T . We show that if T is a weak * Dunford-Pettis operator and F is σ-Dedekind complete, then S itself is weak * Dunford-Pettis.2010 Mathematics subject classification: primary 46B42; secondary 46B50, 47B65.

On positive almost weak* Dunford–Pettis operators

“…Recall also that an operator T : E → X is said to be almost Dunford-Pettis if T maps disjoint weakly null sequences of E to norm null sequences of X . We refer to [1,3,6,7,9] for basic properties of such operators. In [2], the author introduced the notion of almost weak Dunford-Pettis operators on Banach lattices and gave some characterizations of such operators.…”

On the class of weak almost limited operators

“…(2) For every weakly null sequence (x n ) ⊂ E + and every disjoint weak* null sequence (f n ) ⊂ E * we have f n (x n ) → 0. Recently, the authors in [6] demonstrated that if a positive weak* Dunford-Pettis operator T : E → F has its range in σ-Dedekind complete Banach lattice, then every positive operator S : E → F that it dominates (i.e., 0 ≤ S ≤ T ) is also weak* Dunford-Pettis [6, Theorem 3.1]. For the positive weak almost limited operators, the situation still hold when F satisfy the property (d).…”

“…(5) T carries the solid hull of each relatively weakly compact subset of E to an almost limited subset of F . If F has the property (d), we may add:(6) f n (T (x n )) → 0 for every weakly null sequence (x n ) ⊂ E + and every disjoint weak * null sequence(f n ) ⊂ (F * ) + . (7) f n (T (x n )) → 0 for every disjoint weakly null sequence (x n ) ⊂ E + and every disjoint weak * null sequence (f n ) ⊂ (F * ) + .Proof.…”

Some results on various types of compactness of weak* Dunford–Pettis operators on Banach lattices

“…In [16], Grothendieck introduced the notion of Dunford–Pettis operator between Banach spaces, which took a very important place in the mathematical literature, and several of its properties have been studied. In the case of Banach lattices, many authors focus on the various interrelationships between Dunford–Pettis and other types of operators (see, for instance, [1, 4, 5, 6, 10, 11, 17, 20]). Next, a number of related notions have been studied in the literature.…”