Unbounded continuous operators in Banach lattices

Abstract

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce two types of continuous operators between Banach lattices using unbounded absolute weak convergence. We characterize reflexive Banach lattices in terms of these spaces of operators. Furthermore, we investigate whether or not the adjoint of these classes of operators has the corresponding property. In addition, we show that these kinds of operators are norm closed but not order closed. Finally, we show that the notions of an M -weakly operator and a uaw-Dunford-Pettis operator have the same meaning; this extends one of the main results of Erkursun-Ozcan et al. (TJM, 2019).Before to proceed more, let us consider some preliminaries. Suppose E is a Banach lattice. A net (x α ) in E is said to be unbounded absolute weak convergent (uaw-convergent, for short) toBoth convergences are topological. For ample information on these concepts, see [2,4,6].

Some characterizations of L-weakly compact sets using the the unbounded absolutely weakly convergence and applications

“…Also, some characterizations of order continuous and of reflexive Banach lattices are obtained. Several recent papers investigated this concept of convergence have been announced ( [3,4,10,11]).…”