Strict singularity: A lattice approach

Given a Banach lattice E and a Banach space Y  we say that a bounded linear operator T : E → Y  is lattice strictly singular (disjointly strictly singular) if it fails to be invertible on any infinite-dimensional sublattice of E (on the span of any pairwise disjoint sequence in E). This is a survey on the existing answers up to the present day to the following questions: Is every lattice strictly singular operator also disjointly strictly singular? Do lattice strictly singular operators have a vector space structure?