Disjointly non-singular operators: Extensions and local variations

TL;DR

This paper studies disjointly non-singular operators between Banach lattices and Banach spaces, establishing extension properties, local variations, and conditions under which these properties are preserved in ultrapowers and biduals.

Contribution

It introduces a new extension theorem for DNS operators, explores local variations, and shows preservation of DNS properties in ultrapowers and biduals under certain conditions.

Findings

Extensions of DNS operators to $L_1(u)$ spaces are possible.

Ultrapowers of DNS operators remain DNS.

DNS properties are preserved in biduals when certain conditions are met.

Abstract

The disjointly non-singular ($DNS$) operators $T\in L(E,Y)$ 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 $L_1(\mu)$ space, and we show that each of $T\in DNS(E,Y)$ admits an extension $\overline{T}\in DNS(L_1(\mu),PO)$ from which we derive that both $T$ and $T^{**}$ are tauberian operators and that the operator $T^{co}: E^{**}/E\to Y^{**}/Y$ induced by $T^{**}$ is an (into) isomorphism. Also, using a local variation of the notion of $DNS$ operator, we show that the ultrapowers of $T\in DNS(E,Y)$ are also $DNS$ operators. Moreover, when $E$ contains no copies of $c_0$ and admits a weak unit, we show that $T\in DNS(E,Y)$ implies $T^{**}\in DNS(E^{**},Y^{**})$.