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CONTINUITY OF THE CONE SPECTRAL RADIUS.
- Source :
-
Proceedings of the American Mathematical Society . Aug2013, Vol. 141 Issue 8, p2741-2754. 14p. - Publication Year :
- 2013
-
Abstract
- This paper concerns the question whether the cone spectral radius rC(f) of a continuous compact order-preserving homogenous map f : C → 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 < a1 < a2 < a3 < ... not in the cone spectrum, σC(f), and limk→∞ ak = rC(f), then the cone spectral radius is continuous. An example is presented showing that if such a sequence (ak)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 σC(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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 141
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 87765675
- Full Text :
- https://doi.org/10.1090/S0002-9939-2013-11520-0