  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. We extend the Meyer-Nieberg duality theorem to collectively
L-weakly compact sets of operators, study relations between these sets and
collectively almost limited sets, and discuss the domination problem for
collectively compact sets and collectively L-weakly compact sets.
