On the class of weak $$^{\star }$$ Dunford–Pettis operators

On positive almost weak* Dunford–Pettis operators

“…Recall that an operator T from a Banach space X to another Banach space Y is Dunford-Pettis if it maps weakly null sequences of X to norm null sequences of Y , and is weak Dunford-Pettis if f n (T (x n )) → 0 for any weakly null sequence (x n ) in X and any weakly null sequence ( f n ) in Y , and is weak* Dunford-Pettis if f n (T (x n )) → 0 for any weakly null sequence (x n ) in X and any weak* null sequence ( f n ) in Y [6]. 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 .…”

“…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.…”

A note on weak almost limited operators

“…As weak versions of limited operators, several types of operators were recently introduced and studied. An operator T : E → F is -weak* Dunford-Pettis (w*DP) [11], if f n (T x n ) → 0 whenever (x n ) converges weakly to 0 in E and (f n ) converges weak* to 0 in F ′ . -almost limited [18], if ∥T ′ f n ∥ → 0 for every disjoint weak * null sequence (f n ) ⊂ F ′ .…”

Domination by Positive Weak Dunford–pettis Operators on Banach Lattices

“…Recently, H'michane et al [11] introduced the class of weak * Dunford-Pettis operators, and characterised this class of operators and studied some of its properties in [10]. Following H'michane et al [11], we say that a bounded linear operator T : X → Y is a weak * Dunford-Pettis operator whenever x n w − → 0 in X and f n w * − − → 0 in Y * imply f n (T x n ) → 0 or, equivalently, whenever T carries relatively weakly compact subsets of X onto limited subsets of Y [10,Theorem 3.2].…”

“…Aliprantis and Burkinshaw [1] introduced a class of operators related to the Dunford-Pettis operators, the so-called weak Dunford-Pettis operators. A bounded linear operator T : X → Y between Banach spaces is said to be a weak Dunford-Pettis operator whenever x n is a weak * Dunford-Pettis operator whenever x n w − → 0 in X and f n w * − − → 0 in Y * imply f n (T x n ) → 0, or equivalently, whenever T carries relatively weakly compact subsets of X onto limited subsets of Y ( [11,Theorem 3.2]).…”