# On Finite Elements in Lattices of Regular Operators

On Finite Elements in Lattices of Regular Operators Let E and F be vector lattices and $${\mathcal L}^r(E,F)$$ the ordered space of all regular operators, which turns out to be a (Dedekind complete) vector lattice if F is Dedekind complete. We show that every lattice isomorphism from E onto F is a finite element in $${\mathcal L}^r(E,F)$$ , and that if E is an AL-space and F is a Dedekind complete AM-space with an order unit, then each regular operator is a finite element in $${\mathcal L}^r(E,F)$$ . We also investigate the finiteness of finite rank operators in Banach lattices. In particular, we give necessary and sufficient conditions for rank one operators to be finite elements in the vector lattice $${\mathcal L}^r(E,F)$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# On Finite Elements in Lattices of Regular Operators

Positivity, Volume 11 (4) – Sep 26, 2007
12 pages

/lp/springer_journal/on-finite-elements-in-lattices-of-regular-operators-B0RDYPrmxK
Publisher
Springer Journals
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/s11117-007-2007-8
Publisher site
See Article on Publisher Site

### Abstract

Let E and F be vector lattices and $${\mathcal L}^r(E,F)$$ the ordered space of all regular operators, which turns out to be a (Dedekind complete) vector lattice if F is Dedekind complete. We show that every lattice isomorphism from E onto F is a finite element in $${\mathcal L}^r(E,F)$$ , and that if E is an AL-space and F is a Dedekind complete AM-space with an order unit, then each regular operator is a finite element in $${\mathcal L}^r(E,F)$$ . We also investigate the finiteness of finite rank operators in Banach lattices. In particular, we give necessary and sufficient conditions for rank one operators to be finite elements in the vector lattice $${\mathcal L}^r(E,F)$$ .

### Journal

PositivitySpringer Journals

Published: Sep 26, 2007

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