Lemmens, Bas and Nussbaum, Roger
(2013)
*
Continuity of the cone spectral radius.
*
Proceedings of the American Mathematical Society,
141
(8).
pp. 2741-2754.
ISSN 0002-9939.
(doi:10.1090/S0002-9939-2013-11520-0 )
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Official URL http://dx.doi.org/10.1090/S0002-9939-2013-11520-0 |

## Abstract

This paper concerns the question whether the cone spectral radius of a continuous compact order-preserving homogenous map on a closed cone in Banach space depends continuously on the map. Using the fixed point index we show that if there exist points not in the cone spectrum arbitrarily close to the cone spectral radius, then the cone spectral radius is continuous. An example is presented showing that continuity may fail, if this condition does not hold. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that for each polyhedral cone with m faces, the cone spectrum contains at most m-1 elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each non-polyhedral cone, there exist maps whose cone spectrum contains a countably infinite number of distinct points.

Item Type: | Article |
---|---|

Uncontrolled keywords: | Cone spectral radius, continuity, cone spectrum, nonlinear cone maps, fixed point index |

Subjects: | Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus |

Divisions: | Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science |

Depositing User: | Bas Lemmens |

Date Deposited: | 22 Nov 2011 10:09 UTC |

Last Modified: | 07 Jun 2016 14:14 UTC |

Resource URI: | https://kar.kent.ac.uk/id/eprint/28465 (The current URI for this page, for reference purposes) |

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