  Suppose $E$ is a Banach lattice. Recently, there have been some motivating
contexts regarding the known Banach-Saks property and the Grothendieck property
from an order point of view. In this paper, we establish these results for
operators that enjoy different types considered for the Banach-Saks property as
well as for different notions related to the Grothendieck property. In
particular, beside other results, we characterize order continuity and
reflexivity of Banach lattices in terms of the corresponding bounded operators
defined on them.
