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Relatively central operators on Archimedean vector lattices II

Part of: General theory of linear operators Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

P. T. N. McPolin  and
A. W. Wickstead
P. T. N. McPolin
Affiliation:
St. Joseph's College of Education, Belfast BT11 9GA, Northern Ireland
A. W. Wickstead
Affiliation:
Department of Pure Mathematics, The Queen's University of Belfast, Belfast BT7 INN, Northern Ireland
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Abstract

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We continue the study of operators from an Archimedean vector lattice E into a cofinal sublattice H which have the property that there is λ > 0 such that if x ∈ E, h ∈ H and |x|≤|h|, then |Tx| ≤ λ|h|. The collection Z(E|H) of all of those operators forms an algebra under composition. We investigate the relationship between the properties of having an identity, being Abelin and being semi-simple for such-algebras, culminating in a proof that they are equivalent if H is Dedekind complete. We also study various for such an operator T, showing that, apart from 0, its spectrum relative to Z(E|H) is the same as that of T|H relative to Z(H) and that of T relative to ℒ(E) (Provided E is a Banach lattice and H is closed).

MSC classification

Secondary: 47A10: Spectrum, resolvent 46A40: Ordered topological linear spaces, vector lattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 3 , June 1986 , pp. 287 - 298
Copyright
Copyright © Australian Mathematical Society 1986

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