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Continuity of the cone spectral radius

Authors: Bas Lemmens and Roger Nussbaum
Journal: Proc. Amer. Math. Soc. 141 (2013), 2741-2754
MSC (2010): Primary 47H07; Secondary 47H10, 47H14
Published electronically: April 8, 2013
MathSciNet review: 3056564
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Abstract: This paper concerns the question whether the cone spectral radius $ r_C(f)$ of a continuous compact order-preserving homogenous map $ f\colon C\to C$ on a closed cone $ C$ in Banach space $ X$ depends continuously on the map. Using the fixed point index we show that if there exists $ 0<a_1<a_2<a_3<\ldots $ not in the cone spectrum, $ \sigma _C(f)$, and $ \lim _{k\to \infty } a_k = r_C(f)$, then the cone spectral radius is continuous. An example is presented showing that if such a sequence $ (a_k)_k$ does not exist, continuity may fail. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that if $ C$ is a polyhedral cone with $ m$ faces, then $ \sigma _C(f)$ contains at most $ m-1$ elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each nonpolyhedral cone there exist maps whose cone spectrum is infinite.

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Additional Information

Bas Lemmens
Affiliation: School of Mathematics, Statistics & Actuarial Science, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom

Roger Nussbaum
Affiliation: Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

Keywords: Cone spectral radius, continuity, cone spectrum, nonlinear cone maps, fixed point index
Received by editor(s): July 22, 2011
Received by editor(s) in revised form: October 29, 2011
Published electronically: April 8, 2013
Additional Notes: The second author was partially supported by NSF DMS-0701171
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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