Unbounded absolutely weak Dunford–Pettis operators

Abstract

In the present article, we expose various properties of unbounded absolutely weak Dunford-Pettis and unbounded absolutely weak compact operators on a Banach lattice E . In addition to their topological and lattice properties, we investigate relationships between M -weakly compact operators, L -weakly compact operators, and order weakly compact operators with unbounded absolutely weak Dunford-Pettis operators. We show that the square of any positive uaw -Dunford-Pettis ( M -weakly compact) operator on an order continuous Banach lattice is compact. Many examples are given to illustrate the essential conditions.

Unbounded order-norm continuous and unbounded norm continuous operator

“…Let E and F be two Banach lattices. An operator T : E → F, is said to be unbounded absolute weak Dunford-Pettis (or, uaw-Dunford-Pettis for short) if for every norm bounded sequence (x n ) in E, x n uaw − − → 0 in E implies Tx n → 0 in F. This class of operators has been introduced in [7]. If E is a Banach lattice and X is a Banach space.…”

“…1. (a) By Proposition 2.6 of [7] and Theorem 18 of [21], a continuous operator T between two Banach lattices E and F is M-weakly compact iff it is uaw-Dunford-Pettis iff for each norm bounded uo-null sequence (x n ) ⊆ E, T(x n )…”

“…With notice Example 6 from [7], the inclusion mapping 2 → ∞ is not uaw-Dunford-Pettis, and so by proceeding proposition is not un-continuous.…”

“…Remark 2.8. A continuous operator T between two Banach lattices E and F is σ-uon-continuous iff it is M-weakly compact iff it is uaw-Dunford-Pettis (see [7,Proposition 2.6] and [21,Theorem 18]). for all u ∈ E + .…”

Unbounded continuous operators in Banach lattices

“…In particular, unbounded absolute weak convergence as a weak version of unbounded norm convergence and also as an unbounded version of weak convergence has been investigated in [6]. Furthermore, unbounded absolute weak Dunford-Pettis operators ( uaw-Dunford-Pettis operators, in brief), as an unbounded version of Dunford-Pettis operators, have been considered recently in [3]. Now, we consider the following observations as unbounded versions of continuous operators.…”