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Intermediate and extrapolated spaces for bi-continuous operator semigroups

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Abstract

We discuss the construction of the entire Sobolev (Hölder) scale for non-densely defined operators with rays of minimal growth on a Banach space. In particular, we give a construction for extrapolation- and Favard spaces of generators of (bi-continuous) semigroups, or which is essentially the same, Hille–Yosida operators on Saks spaces.

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References

  1. M. Adler, M. Bombieri, and K.-J. Engel. On perturbations of generators of \({C}_{0}\)-semigroups. Abstr. Appl. Anal., 2014:13 pages, 2014.

  2. J. Alber. On implemented semigroups. Semigroup Forum, 63(3):371–386, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Amann. Parabolic evolution equations in interpolation and extrapolation spaces. J. Funct. Anal., 78(2):233–270, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Arendt, O. El-Mennaoui, and V. Kéyantuo. Local integrated semigroups: evolution with jumps of regularity. J. Math. Anal. Appl., 186(2):572–595, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Cerrai. A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum, 49(3):349–367, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. B. Cooper. Saks spaces and applications to functional analysis, volume 139 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, second edition, 1987. Notas de Matemática [Mathematical Notes], 116.

  7. G. Da Prato and P. Grisvard. Maximal regularity for evolution equations by interpolation and extrapolation. Journal of Functional Analysis, 58(2):107 – 124, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Da Prato and P. Grisvard. On extrapolation spaces. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, 72(6):330–332, 1982.

  9. W. Desch and W. Schappacher. On relatively bounded perturbations of linear \({C}_0\)-semigroups. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 11:327–341, 1984.

  10. K.-J. Engel. Spectral theory and generator property for one-sided coupled operator matrices. Semigroup Forum, 58(2):267–295, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  11. K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, 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.

  12. J. Esterle. Mittag-Leffler methods in the theory of Banach algebras and a new approach to Michael’s problem. In Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983), volume 32 of Contemp. Math., pages 107–129. Amer. Math. Soc., Providence, RI, 1984.

  13. B. Farkas. Perturbations of Bi-Continuous Semigroups. PhD thesis, Eötvös Loránd University, 2003.

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

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Greiner. Perturbing the boundary conditions of a generator. Houston J. Math., 13:213–229, 1987.

    MathSciNet  MATH  Google Scholar 

  16. M. Haase. Operator-valued \({H}^{\infty }\)-calculus in inter- and extrapolation spaces. Integral Equations and Operator Theory, 56(2):197–228, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Hadd, R. Manzo, and A. Rhandi. Unbounded perturbations of the generator domain. Discrete Contin. Dyn. Syst., 35(2):703–723, 2015.

    MathSciNet  MATH  Google Scholar 

  18. B. Jacob, S.-A. Wegner, and J. Wintermayr. Desch–Schappacher perturbation of one-parameter semigroups on locally convex spaces. Math. Nachr., 288(8-9):925–934, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Kraaij. Strongly continuous and locally equi-continuous semigroups on locally convex spaces. Semigroup Forum, 92(1):158–185, Feb 2016.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Kunze. Continuity and equicontinuity of semigroups on norming dual pairs. Semigroup Forum, 79(3):540, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Lorenzi and M. Bertoldi. Analytical methods for Markov semigroups. Monographs and Research Notes in Mathematics. Taylor & Francis, 2006.

  23. A. Lunardi. Interpolation theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, second edition, 2009.

  24. P. Magal and S. Ruan. Theory and applications of abstract semilinear Cauchy problems. Springer International Publishing. 2018. https://doi.org/10.1007/978-3-030-01506-0.

  25. R. Nagel. Sobolev spaces and semigroups. Semesterbericht Funktionalanalysis, 1983.

  26. R. Nagel. Extrapolation spaces for semigroups. RIMS Kôkyûroku, 1009:181–191, 1997.

    Google Scholar 

  27. R. Nagel, G. Nickel, and S. Romanelli. Identification of extrapolation spaces for unbounded operators. Quaest. Math., 19(1-2):83–100, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. Nagel and E. Sinestrari. Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators. In Functional analysis (Essen, 1991), volume 150 of Lecture Notes in Pure and Appl. Math., pages 51–70. Dekker, New York, 1994.

  29. R. Nagel and E. Sinestrari. Extrapolation spaces and minimal regularity for evolution equations. J. Evol. Equ., 6(2):287–303, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Priola. On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math., 136(3):271–295, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  31. E. Sinestrari. Interpolation and extrapolation spaces in evolution equations. In J. Cea, D. Chenais, G. Geymonat, and J. L. Lions, editors, Partial Differential Equations and Functional Analysis: In Memory of Pierre Grisvard, pages 235–254. Birkhäuser Boston, Boston, MA, 1996.

  32. O. J. Staffans and G. Weiss. Transfer functions of regular linear systems part III: Inversions and duality. Integral Equations and Operator Theory, 49(4):517–558, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. van Neerven. The Adjoint of a Semigroup of Linear Operators. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1992.

    Book  Google Scholar 

  34. T. Walter. Störungstheorie von Generatoren und Favardklassen. Semesterbericht Funktionalanalysis Tübingen, 1986.

  35. S.-A. Wegner. Universal extrapolation spaces for \(C_0\)-semigroups. Annali dell ’Universita’ di Ferrara, 60(2):447–463, 2014.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are indebted to Rainer Nagel and Sven-Ake Wegner for motivating discussions.

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  1. University of Wuppertal, School of Mathematics and Natural Sciences, Gaussstrasse 20, 42119, Wuppertal, Germany

    Christian Budde & Bálint Farkas

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  1. Christian Budde
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  2. Bálint Farkas
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Correspondence to Bálint Farkas.

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Budde, C., Farkas, B. Intermediate and extrapolated spaces for bi-continuous operator semigroups. J. Evol. Equ. 19, 321–359 (2019). https://doi.org/10.1007/s00028-018-0477-8

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