Disjointly non-singular operators and various topologies on Banach lattices

We continue the study of dispersed subspaces and disjointly non-singular (DNS) operators on Banach lattices using topological methods. In particular, we provide a simple proof of the fact that in an order continuous Banach lattice an operator is DNS if and only if it is $n$-DNS, for some $n\in\mathbb{N}$. We characterize Banach lattices with order continuous dual in terms of dispersed subspaces and absolute weak topology. We also connect these topics with the recently launched study of phase retrieval in Banach lattices.