On some vector lattices of operators and their finite elements - Hahn, Norbert; Hahn, Siegfried; Weber, Martin R. (Positivity : an international journal devoted to the theory and applications of positivity in analysis, 2009)
If the vector space of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for to be a vector lattice. and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space or, in the case that is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM -compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices , where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E , F and are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E ′ and F , are finite elements in . The paper contains also some generalization of results obtained for the case in [10].