Continuity of the cone spectral radius
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- by Bas Lemmens and Roger Nussbaum PDF
- Proc. Amer. Math. Soc. 141 (2013), 2741-2754 Request permission
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.References
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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
- Email: B.Lemmens@kent.ac.uk
- Roger Nussbaum
- Affiliation: Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
- MR Author ID: 132680
- Email: nussbaum@math.rutgers.edu
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2741-2754
- MSC (2010): Primary 47H07; Secondary 47H10, 47H14
- DOI: https://doi.org/10.1090/S0002-9939-2013-11520-0
- MathSciNet review: 3056564