  Recently, the functionals different types of unbounded convergences (uo, un,
uaw, uaw*) in Banach lattices were studied. In this paper, we study the
continuous operators with respect to unbounded convergences. We first
investigate the approximation property of continuous operators for unbounded
convergence. Then we show some characterizations of the continuity of the
continuous operators for uo, un, uaw and uaw*-convergence. Based on these
results, we discuss the order-weakly compact operators on Banach lattices. Some
related results are obtained as well.
