  Motivated by the equivalent definition of a continuous operator between
Banach spaces in terms of weakly null nets, we introduce unbounded continuous
operators by replacing weak convergence with the unbounded absolutely weak
convergence ( $uaw$-convergence) in the definition of a continuous operator
between Banach lattices. We characterize order continuous Banach lattices and
reflexive Banach lattices in terms of these spaces of operators. Moreover,
motivated by characterizing of a reflexive Banach lattice in terms of unbounded
absolutely weakly Cauchy sequences, we consider pre-unbounded operators between
Banach lattices which maps $uaw$-Cauchy sequences to weakly ( $uaw$- or norm)
convergent sequences. This allows us to characterize $KB$-spaces and reflexive
spaces in terms of these operators, too. Furthermore, we consider the unbounded
Banach-Saks property as an unbounded version of the weak Banach-Saks property.
There are many considerable relations between spaces possessing the unbounded
Banach-Saks property with spaces fulfilled by different types of the known
Banach-Saks property. In particular, we characterize order continuous Banach
lattices in terms of these relations, as well.
