On some vector lattices of operators and their finite elements

On some vector lattices of operators and their finite elements If the vector space $${\mathcal{L}}^r(E, F)$$ 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 $${\mathcal{L}}^r(E, F)$$ to be a vector lattice. $${\mathcal{L}}^r(E, F)$$ and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces $${\mathcal{V}}(E, F)$$ of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space $${\mathcal{L}}^r(E, F)$$ or, in the case that $${\mathcal{L}}^r(E, F)$$ 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 $${\mathcal{V}}(E, F)$$ , 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 $${\mathcal{V}}(E, F)$$ 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 $${\mathcal{V}}(E,F)$$ . The paper contains also some generalization of results obtained for the case $${\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)$$ in [10]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On some vector lattices of operators and their finite elements

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Publisher
Birkhäuser-Verlag
Copyright
Copyright © 2008 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2175-1
Publisher site
See Article on Publisher Site

Abstract

If the vector space $${\mathcal{L}}^r(E, F)$$ 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 $${\mathcal{L}}^r(E, F)$$ to be a vector lattice. $${\mathcal{L}}^r(E, F)$$ and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces $${\mathcal{V}}(E, F)$$ of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space $${\mathcal{L}}^r(E, F)$$ or, in the case that $${\mathcal{L}}^r(E, F)$$ 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 $${\mathcal{V}}(E, F)$$ , 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 $${\mathcal{V}}(E, F)$$ 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 $${\mathcal{V}}(E,F)$$ . The paper contains also some generalization of results obtained for the case $${\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)$$ in [10].

Journal

PositivitySpringer Journals

Published: Aug 9, 2008

References

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