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Semigroup Forum

, Volume 96, Issue 2, pp 333–347 | Cite as

Perturbations of positive semigroups on AM-spaces

  • András Bátkai
  • Birgit Jacob
  • Jürgen Voigt
  • Jens Wintermayr
Research Article
First Online:
Received:
Accepted:

Abstract

We consider positive perturbations of positive semigroups on AM-spaces and prove a result which is the dual counterpart of a famous perturbation result of Desch in AL-spaces. As an application we present unbounded perturbations of the shift semigroup.

Keywords

Strongly continuous semigroup Perturbation Positive operators 
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Notes

Acknowledgements

The authors are grateful to Bálint Farkas, Sven-Ake Wegner and Hans Zwart for fruitful discussions and helpful comments. The first author gratefully acknowledges financial support by the German Academic Exchange Service (DAAD).

References

  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985)Google Scholar
  2. 2.
    Arendt, W.: Resolvent positive operators. Proc. London Math. Soc. 54, 321–349 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bátkai, A., Kramer-Fijavž, M., Rhandi, A.: Positive Operator Semigroups: from Finite to Infinite Dimensions. Birkhäuser-Verlag, Basel (2016)Google Scholar
  4. 4.
    Batty, C.J.K., Robinson, D.W.: Positive one-parameter semigroups on ordered Banach spaces. Acta Appl. Math. 2, 221–296 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banasiak, J., Arlotti, L.: Perturbation of Positive Semigroups with Applications. Springer, London (2006)zbMATHGoogle Scholar
  6. 6.
    Davies, E.B.: One-Parameter Semigroups. Academic Press, New York (1980)zbMATHGoogle Scholar
  7. 7.
    Desch, W.: Perturbations of positive semigroups in AL-spaces, preprint (1988)Google Scholar
  8. 8.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)zbMATHGoogle Scholar
  9. 9.
    Goldstein, J.A.: Semigroups of Operators and Applications. Oxford University Press, New York (1985)zbMATHGoogle Scholar
  10. 10.
    Jacob, B., Zwart, H.J.: Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. Birkhäuser-Verlag, Basel (2012)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1980)zbMATHGoogle Scholar
  12. 12.
    Nagel, R. (ed.): One-Parameter Semigroups of Positive Operators. Springer, Berlin (1986)zbMATHGoogle Scholar
  13. 13.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  14. 14.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  15. 15.
    Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1980)zbMATHGoogle Scholar
  16. 16.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)zbMATHGoogle Scholar
  17. 17.
    Tucsnak, M., Weiss, G.: Observation and Control for Operator Semigroups. Birkhäuser-Verlag, Basel (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Voigt, J.: On resolvent positive operators and positive \(C_0\)-semigroups on AL-spaces. Semigroup Forum 38, 263–266 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • András Bátkai
    • 1
    • 2
  • Birgit Jacob
    • 1
  • Jürgen Voigt
    • 3
  • Jens Wintermayr
    • 1
  1. 1.School of Mathematics and Natural SciencesBergische Universität WuppertalWuppertalGermany
  2. 2.Pädagogische Hochschule VorarlbergFeldkirchAustria
  3. 3.Fachrichtung MathematikTechnische Universität DresdenDresdenGermany

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