The Library
Extension of orderpreserving maps on a cone
Tools
UNSPECIFIED. (2003) Extension of orderpreserving maps on a cone. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION AMATHEMATICS, 133 (Part 1). pp. 3559. ISSN 03082105
Full text not available from this repository.Abstract
We examine the problem of extending, in a natural way, orderpreserving maps that are defined on the interior of a closed cone K1 (taking values in another closed cone K2) to the whole of K1.
We give conditions, in considerable generality (for cones in both finite and infinitedimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semicontinuous. Maps f defined on the interior of the nonnegative cone K in RN, which are both homogeneous of degree 1 and order preserving, are nonexpanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous orderpreserving 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 operatorvalued 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 AMATHEMATICS  
Publisher:  ROYAL SOC EDINBURGH  
ISSN:  03082105  
Official Date:  2003  
Dates: 


Volume:  133  
Number:  Part 1  
Number of Pages:  25  
Page Range:  pp. 3559  
Publication Status:  Published  
URI:  http://wrap.warwick.ac.uk/id/eprint/9886 
Data sourced from Thomson Reuters' Web of Knowledge
Request changes or add full text files to a record
Actions (login required)
View Item 