Continuity of the cone spectral radius

Bas Lemmens, Roger Nussbaum

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

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 < aC1 < 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.

Original languageEnglish (US)
Pages (from-to)2741-2754
Number of pages14
JournalProceedings of the American Mathematical Society
Volume141
Issue number8
DOIs
StatePublished - Aug 2013

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Mathematics(all)

Keywords

  • Cone spectral radius
  • Cone spectrum
  • Continuity
  • Fixed point index
  • Nonlinear cone maps

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