Unbounded continuous operators in Banach lattices

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. Moreover, we establish how these classes of continuous operators are connected to the well-known classes of $M$-weakly compact operators and $L$-weakly compact operators. 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).