A set of bounded linear operators from a Banach space to a Banach lattice is collectively \(L\) -weakly compact whenever union of images of the unit ball is \(L\) -weakly compact. In the present note the Meyer-Nieberg duality theorem is extended to collectively \(L\) -weakly compact sets of operators, the relationship between collectively \(L\) -weakly compact sets and collectively almost limited sets is investigated, and the domination problem for collectively compact and collectively \(L\) -weakly compact sets is studied.
