Unbounded absolutely weak Dunford–Pettis operators

Nazife Erkurşun Özcan,Niyazi Anıl Gezer,Omid Zabeti

doi:10.3906/mat-1904-27

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.

Highlights

Introduction and preliminariesThe concept of unbounded order convergence under the name of individual convergence was first considered in [13] and “ uo -convergence” was initially proposed in [6]
We reveal the relationships between uaw -Dunford–Pettis operators, unbounded absolutely weak compact operators, M -weakly compact operators, L-weakly compact operators, and o-weakly compact operators
We show that in the case of a uaw -Dunford–Pettis operator, the situation is different

Summary

Introduction

Introduction and preliminariesThe concept of unbounded order convergence under the name of individual convergence was first considered in [13] and “ uo -convergence” was initially proposed in [6]. As one of main consequences, we deduce that the square of a positive uaw -Dunford–Pettis ( M -weakly compact) operator on an order continuous Banach lattice is compact. Proof Suppose T is Dunford–Pettis and xn is a norm bounded sequence in E , which is uaw -convergent to zero. Proof If T and S are two uaw -Dunford–Pettis operators and xn is a norm bounded sequence satisfying xn −u−a−→w 0 ||T S(xn)|| −→ 0 and ||(T + S)xn|| −→ 0 .

Results

Conclusion