UNSPECIFIED (2003) Extension of order-preserving maps on a cone. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 133 (Part 1). pp. 35-59. ISSN 0308-2105

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Abstract

We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone K-1 (taking values in another closed cone K-2) to the whole of K-1. We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous. Maps f defined on the interior of the non-negative cone K in R-N, which are both homogeneous of degree 1 and order preserving, are non-expanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that such an extension must have at least one eigenvector in K - {0}. In the case where the cycle time chi(f) of the original map does not exist, such eigenvectors must lie in partial derivativeK - {0}. We conclude with some discussions and applications to operator-valued means. We also extend our results to an 'intermediate' situation, which arises in some important application areas, particularly in the construction of diffusions on certain fractals via maps defined on the interior of cones of Dirichlet forms.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Journal or Publication Title:	PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
Publisher:	ROYAL SOC EDINBURGH
ISSN:	0308-2105
Date:	2003
Volume:	133
Number:	Part 1
Number of Pages:	25
Page Range:	pp. 35-59
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/9886

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