On some vector lattices of operators and their finite elements

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...

Full description

Saved in:
Bibliographic Details
Published in:Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 13; no. 1; pp. 145 - 163
Main Authors: Hahn, Norbert, Hahn, Siegfried, Weber, Martin R.
Format: Journal Article
Language:English
Published: Basel Birkhäuser-Verlag 01.02.2009
Springer
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics. Ordered structures
Econometrics
Exact sciences and technology
Fourier Analysis
Functional analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Operator Theory
Order, lattices, ordered algebraic structures
Potential Theory
Sciences and techniques of general use
Vector space
regular operator
Banach lattice
finite element
Vector lattice
46B42
47B65
modulus of an operator
compact operator
AM-compact operator
47B60
47B07
finite rank operator
weakly compact operator
Dunford-Pettis operator
Rank
Linear operator
Ordered space
Finite element method
Compact operator
Sufficient condition
Vector subspace
Vector space
F space
Mathematical analysis
Completeness
Vector
Finite element
Positive operator
Sublattice
ISSN:1385-1292, 1572-9281
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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].
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:1385-1292
1572-9281
DOI:10.1007/s11117-008-2175-1