Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology

In this article we investigate disjointly non-singular (DNS) operators. Following [9] say that an operator T from a Banach lattice F into space E is DNS, if no restriction of to subspace generated by disjoint sequence strictly singular. We partially answer question showing class operators forms open subset L(F,E) as soon order continuous. Moreover, show in case DNS and only the norm topology minimal which simultaneously stronger than unbounded map (we “complements” F). Since plays similar role category lattices upper semi-Fredholm play spaces, indeed uncover characterization latter operators, but time they have complement weak topology.