# 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

Positivity, Volume 13 (1) – Aug 9, 2008
19 pages

Loading next page...

/lp/springer_journal/on-some-vector-lattices-of-operators-and-their-finite-elements-0MV20MoQLU
Publisher
Springer Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off