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 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 .
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
