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Positivity

, Volume 17, Issue 3, pp 875–898 | Cite as

On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators

  • Moritz GerlachEmail author
Article

Abstract

Given a positive, irreducible and bounded \(C_0\)-semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or kernel operator. For a discrete semigroup, i.e. for powers of a single operator \(T\), we show that the point spectrum of some power \(T^k\) intersects the unit circle at most in \(1\). As a consequence, we obtain a sufficient condition for strong convergence of the \(C_0\)-semigroup and for a subsequence of the powers of \(T\), respectively.

Keywords

Peripheral point spectrum Irreducible Harris operator  Compact operator Asymptotic behavior 

Mathematics Subject Classification (2000)

Primary 47A11 Secondary 47B65 47G10 47D06 

Notes

Acknowledgments

The author was supported by the graduate school Mathematical Analysis of Evolution, Information and Complexity during the work on this article and he would like to thank Wolfgang Arendt for many helpful discussions and the anonymous referee for his/her constructive comments.

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institute of Applied AnalysisUniversity of UlmUlmGermany

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