Continuity of the cone spectral radius

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. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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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
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Depositing User: Bas Lemmens
Date Deposited: 22 Nov 2011 10:09
Last Modified: 15 Jul 2014 14:20
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