Disjointly non-singular operators: Extensions and local variations

Abstract

The disjointly non-singular (DN-S) operators T∈L(E, Y) from a Banach lattice Eto a Banach space Yare those operators which are strictly singular in no closed subspace generated by a disjoint sequence of non-zero vectors. When Eis order continuous with a weak unit, Ecan be represented as a dense ideal in some L1(μ) space, and we show that each T∈DN-S(E, Y) admits an extension T∈DN-S(L1(μ), PO), where POis certain Banach space, from which we derive that both Tand T∗∗are tauberian operators and that the operator Tco: E∗∗/E→Y∗∗/Y induced by T∗∗is an (into) isomorphism. Also, using a local variation of the notion of DN-S operator, we show that the ultrapowers of T∈DN-S(E, Y) are also DN-Soperators. Moreover, when Econtains no copies of c0and admits a weak unit, we show that T∈ DN-S(E, Y) implies T∗∗∈ DN-S(E∗∗, Y∗∗).