Disjointly non-singular operators on Banach lattices

Abstract

An operator T from a Banach lattice E into a Banach space is disjointly non-singular (DN-S, for short) if no restriction of T to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for DN-S operators, including a perturbative characterization. For E=Lp (1<p<∞) we improve the results, and we show that the DN-S operators have a different behavior in the cases p=2 and p≠2. As an application we prove that the strongly embedded subspaces of Lp form an open subset in the set of all closed subspaces.