Weak limited sets and operators on Banach lattices

In this paper, we prove that an operator T : E -> F , between two Banach lattices, maps order intervals onto weak limited sets if and only if the modulus |ST| exists and is Dunford-Pettis for every Dunford-Pettis operator S : F -> c0. Next, we establish that a Banach lattice E does not contain any isomorphic copy of l(1) if and only if the order intervals of E are weak limited and the norm of E ' is order continuous. We also investigate the domination problem of the class of weak limited operators.