Abstract
The disjointly non-singular (DN-S) operators T∈L(E,Y) from a Banach lattice E to a Banach space Y are those operators which are strictly singular in no closed subspace generated by a disjoint sequence of non-zero vectors. When E is order continuous with a weak unit, E can 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 PO is certain Banach space, from which we derive that both T and 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-S operators. Moreover, when E contains no copies of c0 and admits a weak unit, we show that T∈DN-S(E,Y) implies T⁎⁎∈DN-S(E⁎⁎,Y⁎⁎).
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