  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).
