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Positivity

, Volume 20, Issue 2, pp 307–336 | Cite as

On the peripheral spectrum of positive operators

  • Jochen Glück
Article

Abstract

This paper contributes to the analysis of the peripheral (point) spectrum of positive linear operators on Banach lattices. We show that, under appropriate growth and regularity conditions, the peripheral point spectrum of a positive operator is cyclic and that the corresponding eigenspaces fulfil a certain dimension estimate. A couple of examples demonstrates that some of our theorems are optimal. Our results on the peripheral point spectrum are then used to prove a sufficient condition for the peripheral spectrum of a positive operator to be cyclic; this generalizes theorems of Lotz and Scheffold.

Keywords

Positive operator Perron-Frobenius theory Peripheral spectrum Peripheral point spectrum Cyclic Growth condition 

Mathematics Subject Classification

47B65 47A10 

Notes

Acknowledgments

I would like to thank Manuel Bernhard and Manfred Sauter for their help in the construction of Example 7.7; moreover, Manfred Sauter assisted me with the proof of Lemma A.4. My thanks also go to Rainer Nagel who suggested the investigation of weakly almost periodic operators in the context of Theorem 5.5.

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Institute of Applied AnalysisUlm UniversityUlmGermany

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