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

Abstract

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