The Regularity Of Operators On An Extended α-normed Separable Banach Lattice

This article is mainly divided into three parts. In the first part, we have some advanced argument on the properties of the A-topology for an Archimedean Riesz space E and the properties of the topological space E equipped with the A-topology. Two main results are obtained as follow (1) If one of the following conditions holds for an Archimedean Riesz space E and a non-empty subset A c E, then the topological space E equipped the A-topology is a Hausdorff space: (a) The subset {|a| : a ∈A} has an upper bound in E; (b) E has a strong unit; (c) In the case of E = C(X), X is a locally compact Hausdorff space. (2) If / is an ideal in an Archimedean Riesz space E and I(?) A≠φ, then the ideal I is both A-open and A-closed.Based on the properties of a-topology and Zorn's lemma, the main result in the second part says that if H is a separable extended a-normed Banach lattice and Y is a Banach lattice with the Cantor property, then every order bounded linear operator T : H →Y is regular.The third part of this article solves the extension of positive operators preserving order, under the domination of a sublinear operator. Let X be a separable Banach lattice and Y be a Banach lattice with the Cantor property. P : X →Y+ is an absolute and continuous sublinear operator and T : X →Y is a continuous positive linear operator. The main result means that if X₀ is a vector subspace of X and V : X₀ →Y is a continuous linear operator satisfying V ≥T|X₀ and V(x) ≤P(x) for all x ∈X₀, then there exists a continuous extension V of V such that V ≥T on X and V (x) ≤P(x) for every x € X.