Dunford-Pettis and Compact Operators Based on Unbounded Absolute Weak Convergence

Abstract

In this paper, using the concept of unbounded absolute weak convergence ($uaw$-convergence, for short) in a Banach lattice, we define two classes of continuous operators, named $uaw$-Dunford-Pettis and $uaw$-compact operators. We investigate some properties and relations between them. In particular, we consider some hypotheses on domain or range spaces of operators such that the adjoint or the modulus of a $uaw$-Dunford-Pettis or $uaw$-compact operator inherits a similar property. In addition, we look into some connections between compact operators, weakly compact operators, and Dunford-Pettis ones with $uaw$-versions of these operators. Moreover, we examine some relations between $uaw$-Dunford-Pettis operators, $M$-weakly compact operators, $L$-weakly compact operators, and $o$-weakly compact ones. As a significant outcome, 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, as well.