On Operators Whose Compactness Properties Are Defined by Order

On Operators whose Compactness Properties are defined by Order by Safak Alpay Abstract. Let E be a Banach lattice. A subset B of E is called order bounded if there exist a, b in E such that a ≤ x ≤ b for each x ∈ B. Considering E in E′′, the bidual of E, a subset B of E is called b-order bounded in E if it is order bounded in the Banach lattice E′′. A bounded linear operator T : E → X is called o-weakly compact if T (B) is relatively weakly compact for each order bounded set B in E. T is called b-weakly compact if T (B) is relatively weakly compact for each b-order bounded subset B in E. T is called an operator of strong type B if T (B(E)) ⊂ X where B(E) is the band generated by E in E′′. Let these spaces be denoted by Wo(E,X),Wb(E,X), and Wst(E,X) respectively and let W (E,X) be the space of weakly compact operators between E and X. In general, we have Let E be a Banach lattice. A subset B of E is called order bounded if there exist a, b in E such that a ≤ x ≤ b for each x ∈ B. Considering E in E′′, the bidual of E, a subset B of E is called b-order bounded in E if it is order bounded in the Banach lattice E′′. A bounded linear operator T : E → X is called o-weakly compact if T (B) is relatively weakly compact for each order bounded set B in E. T is called b-weakly compact if T (B) is relatively weakly compact for each b-order bounded subset B in E. T is called an operator of strong type B if T (B(E)) ⊂ X where B(E) is the band generated by E in E′′. Let these spaces be denoted by Wo(E,X),Wb(E,X), and Wst(E,X) respectively and let W (E,X) be the space of weakly compact operators between E and X. In general, we have W (E,X) ⊆Wst(E,X) ⊆Wb(E,X) ⊆Wo(E,X) between the spaces introduced above where containments may be strict. We study the pair of spaces where equalities hold above and show how the equalities of these spaces help to characterize various properties of E and X. Vector lattices associated with ordered vector spaces by Richard Becker Abstract. Let V be a real, Archimedian ordered, vector space, whose positive cone V + satisfies V = V −V . To V we can associate a Dedekind complete vector lattice W containing V (by abuse of notations), using cuts theory. When V is a vector lattice, W is the Dedekind complete vector lattice associated to V . In the case when V has an order unit the determination of W is already known. Let W0 ⊂W be the vector lattice generated by V . We study W0 in the case where the cone C of all positive linear forms on V separates the elements of V . Moreover, we assume that v ∈ V belongs to V + iff `(v) ≥ 0 for every ` ∈ C. The determination of W0 involves the extreme rays of C. Let Eg(C) be the union of these rays. When C is the closed convex hull of Eg(C) we determine W0 in terms of functions on Eg(C). In any case we determine the cone of all positive linear forms on W0 in terms of conical measures on C, in the sense of G.Choquet. We formulate several problems within the framework of well-capped cones and ice-cream cones. References R.Becker, Vector Lattices Associated with Ordered Vector Spaces, Mediterr. J. Math. 7, (2010), 313-322. Let V be a real, Archimedian ordered, vector space, whose positive cone V + satisfies V = V −V . To V we can associate a Dedekind complete vector lattice W containing V (by abuse of notations), using cuts theory. When V is a vector lattice, W is the Dedekind complete vector lattice associated to V . In the case when V has an order unit the determination of W is already known. Let W0 ⊂W be the vector lattice generated by V . We study W0 in the case where the cone C of all positive linear forms on V separates the elements of V . Moreover, we assume that v ∈ V belongs to V + iff `(v) ≥ 0 for every ` ∈ C. The determination of W0 involves the extreme rays of C. Let Eg(C) be the union of these rays. When C is the closed convex hull of Eg(C) we determine W0 in terms of functions on Eg(C). In any case we determine the cone of all positive linear forms on W0 in terms of conical measures on C, in the sense of G.Choquet. We formulate several problems within the framework of well-capped cones and ice-cream cones. References R.Becker, Vector Lattices Associated with Ordered Vector Spaces, Mediterr. J. Math. 7, (2010), 313-322.