Unbounded Banach–Saks operators and unbounded Grothendieck operators on Banach lattices

Recently, there has been much attention to the ordered structures beyond Banach lattices. Moreover, we have many nice properties in Banach spaces that can be transformed naturally into Banach lattice cases. Therefore, combining these notions with order structure can produce nicer notions, as well. 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. We characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding operator versions related to the Banach–Saks properties. Moreover, we consider different notions related to the Grothendieck property from an ordered attitude; then, we investigate operator versions of these concepts, as well. In particular, beside other results, we characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding bounded linear operators defined on the corresponding Banach lattices, as well.