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On the stability of positive semigroups generated by operator matrices

  • Rainer Nagel
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1422)

Abstract

We consider unbounded operator matrices generating positive semigroups on products of Banach lattices. Generalizing the concept of an M-matrix (see [2]) we characterize the stability of the generated semigroup by simple criteria.

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References

  1. [1]
    A. BELLENI-MORANTE, “Applied Semigroups and Evolution Equations”, Oxford University Press 1979.Google Scholar
  2. [2]
    A. BERMAN AND J. PLEMMONS, “Nonnegative Matrices in the Mathematical Sciences”, Academic Press, New York 1979.zbMATHGoogle Scholar
  3. [3]
    V. CAPASSO AND L. MADDALENA, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biology 13 (1981), 173–184.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    PH. CLEMENT, H. J. A. M. HEIJMANS et al., “One-parameter Semigroups”, CWJ Monographs 5, North Holland, Amsterdam 1987.zbMATHGoogle Scholar
  5. [5]
    M. FIEDLER AND V. PTAK, On matrices with non-positive off-diagonal elements and positive principal minors, Czech. Math. J. 12 (1962), 382–400.MathSciNetzbMATHGoogle Scholar
  6. [6]
    J. A. GOLDSTEIN, “Semigroups of Linear Operators and Applications”, Oxford University Press, New York 1985.zbMATHGoogle Scholar
  7. [7]
    A. GRABOSCH, Translation semigroups and their linearizations on spaces of integrable functions, Trans. Amer. Math. Soc. (to appear)Google Scholar
  8. [8]
    G. GREINER AND R. NAGEL, On the stability of stongly continuous semigroups of positive operators on L 2(μ), Annali Scuola Normale Sup. Pisa 10 (1983), 257–262.MathSciNetzbMATHGoogle Scholar
  9. [9]
    G. GREINER AND R. NAGEL, Growth of cell populations via one-parameter semigroups of positive operators. In: J. A. Goldstein, S. Rosencrans, G. Sod (eds.): “Mathematics Applied to Science”, Academic Press 1988, p. 79–105.Google Scholar
  10. [10]
    H. MINC, “Nonnegative Matrices”, Wiley-Interscience 1988.Google Scholar
  11. [11]
    R. Nagel, Well-posedness and positivity for systems of linear evolution equations, Conferenze del Seminario di Matematica Bari 203 (1985), 1–29.MathSciNetzbMATHGoogle Scholar
  12. [12]
    R. NAGEL (ed.), “One-Parameter Semigroups of Positive Operators”, Lecture Notes Math. 1184, Springer-Verlag 1986.Google Scholar
  13. [13]
    R. NAGEL, Towards a “matrix theory” for unbounded operator matrices, Math. Z. (to appear)Google Scholar
  14. [14]
    A. PAZY, “Semigroups of Linear Operators and Applications to Partial Differential Equations”, Springer-Verlag 1983.Google Scholar
  15. [15]
    H. H. SCHAEFER, “Banach Lattices and Positive Operators”, Springer-Verlag 1974.Google Scholar
  16. [16]
    E. VESENTINI, Semigroups of holomorphic isometries, Adv. Math. 65 (1987), 272–306.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Rainer Nagel
    • 1
  1. 1.Mathematisches Institut der Universität TübingenTübingen

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