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Positive Miyadera–Voigt perturbations of bi-continuous semigroups

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We discuss positive Miyadera–Voigt type perturbations for bi-continuous semigroups on \(\mathrm {AL}\)-spaces with an additional locally convex topology generated by additive seminorms. The main example of such spaces is the space of bounded Borel measures (on a Polish space).

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References

  1. Adler, M., Bombieri, M., Engel, K.-J.: On perturbations of generators of \(C_0\)-semigroups. Abstr. Appl. Anal. Art. ID 213020, 13 (2014)

  2. Arendt, W.: Resolvent positive operators. Proc. Lond. Math. Soc. (3) 54(2), 321–349 (1987)

    Article  MathSciNet  Google Scholar 

  3. Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H.P., Moustakas, U., Nagel, R., Neubrander, F., Schlotterbeck, U.: One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, vol. 1184. Springer, Berlin (1986)

    Book  Google Scholar 

  4. Arendt, W., Rhandi, A.: Perturbation of positive semigroups. Arch. Math. (Basel) 56(2), 107–119 (1991)

    Article  MathSciNet  Google Scholar 

  5. Banasiak, J., Arlotti, L.: Perturbations of Positive Semigroups with Applications. Springer Monographs in Mathematics. Springer, London (2006)

    MATH  Google Scholar 

  6. Bátkai, A., Kramar Fijavž, M., Rhandi, A.: Positive Operator Semigroups, Volume 257 of Operator Theory: Advances and Applications. Birkhäuser/Springer, Cham (2017). (From finite to infinite dimensions, with a foreword by Rainer Nagel and Ulf Schlotterbeck)

  7. Bogachev, V.I.: Differentiable Measures and the Malliavin Calculus, Mathematical Surveys and Monographs, vol. 164. American Mathematical Society, Providence (2010)

    Google Scholar 

  8. Bombieri, M.: On Weiss–Staffans Perturbations of Semigroup Generators. Ph.D. thesis. Eberhard-Karls-Universität Tübingen (2015)

  9. Budde, C.: General Extrapolation Spaces and Perturbations of Bi-continuous Semigroups. Ph.D. thesis. Bergische Universität, Wuppertal (2019)

  10. Budde, C., Farkas, B.: A Desch–Schappacher perturbation theorem for bi-continuous semigroups. Math Nachr. 293, 1053–1073 (2020)

  11. Budde, C., Farkas, B.: Intermediate and extrapolated spaces for bi-continuous operator semigroups. J. Evol. Equ. 19(2), 321–359 (2019)

    Article  MathSciNet  Google Scholar 

  12. Conway, J.B.: A Course in Abstract Analysis, Graduate Studies in Mathematics, vol. 141. American Mathematical Society, Providence (2012)

    Book  Google Scholar 

  13. Cristescu, R.: Vector seminorms, spaces with vector norm, and regular operators. Rev. Roum. Math. Pures Appl. 53(5–6), 407–418 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Desch, W.: Perturbations of positive semigroups in AL-spaces. Preprint (1988)

  15. Desch, W., Schappacher, W.: On relatively bounded perturbations of linear \(C_0\)-semigroups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11(2), 327–341 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Volume 194 of Graduate Texts in Mathematics. Springer, New York (2000). (With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt)

  17. Es-Sarhir, A., Farkas, B.: Positivity of perturbed Ornstein–Uhlenbeck semigroups on \(C_b(H)\). Semigr. Forum 70(2), 208–224 (2005)

    Article  Google Scholar 

  18. Farkas, B.: Perturbations of Bi-continuous Semigroups. Ph.D. thesis. Eötvös Loránd University (2003)

  19. Farkas, B.: Perturbations of bi-continuous semigroups. Stud. Math. 161(2), 147–161 (2004)

    Article  MathSciNet  Google Scholar 

  20. Farkas, B.: Perturbations of bi-continuous semigroups with applications to transition semigroups on \(C_b(H)\). Semigr. Forum 68(1), 87–107 (2004)

    Article  Google Scholar 

  21. Farkas, B.: Adjoint bi-continuous semigroups and semigroups on the space of measures. Czechoslov. Math. J. 61(136), 309–322 (2011)

    Article  MathSciNet  Google Scholar 

  22. Farkas, B., Lorenzi, L.: On a class of hypoelliptic operators with unbounded coefficients in \(\mathbb{R}^N\). Commun. Pure Appl. Anal. 8, 1159 (2009)

    Article  MathSciNet  Google Scholar 

  23. Fomin, S.V.: Differentiable measures in linear spaces. Usehi Mat. Nauk 23(1 (139)), 221–222 (1968)

    MathSciNet  Google Scholar 

  24. Kühnemund, F.: Bi-continuous Semigroups on Spaces with Two Topologies: Theory and Applications. Ph.D. thesis. Eberhard-Karls-Universiät Tübingen (2001)

  25. Kühnemund, F.: A Hille–Yosida theorem for bi-continuous semigroups. Semigr. Forum 67(2), 205–225 (2003)

    Article  MathSciNet  Google Scholar 

  26. Lorenzi, L., Bertoldi, M.: Analytical Methods for Markov Semigroups, Pure and Applied Mathematics (Boca Raton), vol. 283. Chapman & Hall/CRC, Boca Raton (2007)

    MATH  Google Scholar 

  27. Lunardi, A.: Interpolation Theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 2nd edn. Edizioni della Normale, Pisa (2009)

    Google Scholar 

  28. Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. I. North-Holland Publishing Co., Amsterdam (1971)

    MATH  Google Scholar 

  29. Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)

    Book  Google Scholar 

  30. Rudin, W.: Functional Analysis, International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)

    Google Scholar 

  31. Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1971). (Third printing corrected, Graduate Texts in Mathematics, vol. 3)

    Book  Google Scholar 

  32. Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, New York (1974). (Die Grundlehren der mathematischen Wissenschaften, Band 215)

    Book  Google Scholar 

  33. Scirrat, A.K.: Evolution Semigroups for Well-Posed, Non-autonomous Evolution Families. Ph.D. thesis. Louisiana State University and Agricultural and Mechanical College (2016)

  34. Skorohod, A.V.: On the differentiability of measures which correspond to stochastic processes I. Processes with independent increments. Teor. Veroyatnost. i Primenen 2, 417–443 (1957)

    MathSciNet  Google Scholar 

  35. Skorohod, A.V.: Integration in Hilbert Space. Springer, New York (1974). (Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 79)

    Book  Google Scholar 

  36. van Neerven, J.: The Adjoint of a Semigroup of Linear Operators, Lecture Notes in Mathematics, vol. 1529. Springer, Berlin (1992)

    Book  Google Scholar 

  37. Voigt, J.: On the perturbation theory for strongly continuous semigroups. Math. Ann. 229(2), 163–171 (1977)

    Article  MathSciNet  Google Scholar 

  38. Voigt, J.: On resolvent positive operators and positive \(C_0\)-semigroups on \(AL\)-spaces. Semigr. Forum 38(2), 263–266 (1989). Semigroups and differential operators (Oberwolfach, 1988)

    Article  Google Scholar 

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Acknowledgements

The author would like to thank Bálint Farkas for the suggestion of the topic of this paper as well as for the fruitful discussions and the continuous support during writing. Moreover, the author is indebted to Sanne ter Horst for helpful feedback. Furthermore, the author is also very grateful to the referee’s suggestions. They have led to major improvements of the article. Especially, the readability of the article overall has been improved.

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  1. North-West University, School of Mathematical and Statistical Sciences, Potchefstroom Campus, Private Bag X6001-209, Potchefstroom, 2520, South Africa

    Christian Budde

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Correspondence to Christian Budde.

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The author was supported by the DAAD-TKA Project 308019 Coupled systems and innovative time integrators.

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Budde, C. Positive Miyadera–Voigt perturbations of bi-continuous semigroups. Positivity 25, 1107–1129 (2021). https://doi.org/10.1007/s11117-020-00806-1

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