Some characterizations of almost limited operators

Adjoints and second adjoints of almost Dunford-Pettis operators

“…A similar result for the second adjoint T * * is also proved. The results of these sections extend and generalize the known results on the topic, especially those from [3,9,11]. Finally, in Section 5 we prove when the second adjoint of an order weakly compact operator is almost Dunford-Pettis.…”

“…To study the iteration of Theorem 3.4 with itself, recall that a regular operator T : E −→ F between Banach lattices is almost limited if T * takes disjoint weak * -null sequences in F * to norm null sequences in E * (see [9]). Since Dedekind σ-complete Banach lattices have property (d) and T * is defined on a Dedekind complete Banach lattice, from [9, Theorem 3] it follows that T * is almost limited if and only if T * is positively limited.…”

“…In this paper investigate when the adjoint and the second adjoint of an operator are almost Dunford-Pettis. These and related problems were treated in, e.g., [3,9,11].…”

Regularly limited, Grothendieck, and Dunford--Pettis operators

“…In this section, we investigate several modifications of limited operators, introduced recently in [16,23,25,26,30,32,33,38]. A bounded subset B ⊆ E ′ is called: c) an almost L-set (shortly, an a-L-set) if every disjoint w-null sequence (x n ) in E is uniformly null on B (cf.…”

Grothendieck-type subsets of Banach lattices

“…→ 0 for every disjoint weak* null sequence (x ′ n ). This property was introduced by Elbour in [8]. It is easy to see that every Banach lattice with the weak Grothendieck property has the property (d).…”