Skip to main content
SpringerLink
Log in

C-semigroups and strongly continuous semigroups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖[Im(C)]≡‖C −1 x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true.

We construct fractional powers of generators of boundedC-semigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Arendt and H. Kellermann,Integrated solutions of Volterra integro-differential equations and applications, inIntegro-Differential Equations, Proc. Conf. Trento, 1987 (G. Da Prato and Ma. Iannelli eds.), Pitman, London, to appear.

  2. W. Arendt, F. Neubrander and U. Schlotterbeck,Interpolation of semigroups and integrated semigroups, Semesterbericht Funktionalanalysis, Tübingen, Wintersemester, 1988/89.

  3. A.V. Balakrishnan,Fractional powers of closed operators and the semigroups generated by them, Pac. J. Math.10 (1960), 419–437.

    MATH  Google Scholar 

  4. R. Beals,On the abstract Cauchy problem, J. Funct. Anal.10 (1972), 281–299.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Beals,Semigroups and abstract Gevrey spaces, J. Funct. Anal.10 (1972), 300–308.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Bürger,Perturbations of positive semigroups and applications to population genetics, Math. Z.197 (1988), 259–272.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Bürger,An integro-differential equation from population genetics and perturbations of differentiable semigroups in Frechet spaces, preprint (1989).

  8. R. Bürger,Unbounded one-parameter semigroups, Frechet spaces and applications, preprint (1989).

  9. G. Da Prato,Semigruppi regolarizzabili, Ricerche Mat.15 (1966), 223–248.

    MATH  MathSciNet  Google Scholar 

  10. G. Da Prato,R-semigruppi analitici ed equazioni di evoluzione in L p, Ricerche Mat.16 (1967), 223–249.

    Google Scholar 

  11. E.B. Davies,One-Parameter Semigroups, Academic Press, London, 1980.

    MATH  Google Scholar 

  12. E.B. Davies and M.M. Pang,The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc.55 (1987), 181–208.

    Article  MATH  MathSciNet  Google Scholar 

  13. R. deLaubenfels,Powers of generators of holomorphic semigroups, Proc. Amer. Math. Soc.99 (1987), 105–108.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. deLaubenfels,C-semigroups and the Cauchy problem, J. Funct. Anal., to appear.

  15. R. de Laubenfels,Integrated semigroups, C-semigroups and the abstract Cauchy problem, Semigroup Forum41 (1990), 83–95.

    Article  MathSciNet  Google Scholar 

  16. R. deLaubenfels,Entire solutions of the abstract Cauchy problem, Semigroup Forum42 (1991), 83–105.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. deLaubenfels,Existence and uniqueness families for the abstract Cauchy problem, Proc. London Math. Soc., to appear.

  18. R. deLaubenfels,Bounded commuting multiplicative perturbations of generators of strongly continuous groups, Houston Math. J., to appear.

  19. R. deLaubenfels,Holomorphic C-existence families, Tokyo J. Math., to appear.

  20. H.O. Fattorini,The Abstract Cauchy Problem, Addison-Wesley, Reading, Mass., 1983.

    Google Scholar 

  21. J.A. Goldstein,Semigroups of Operators and Applications, Oxford, New York, 1985.

    MATH  Google Scholar 

  22. J. A. Goldstein, R. deLaubenfels and J. T. Sandefur,Regularized semigroups, iterated Cauchy problem, and equipartition of energy, Monat. Math., to appear.

  23. K. Ha,Spectral mapping theorems for exponentially bounded C-semigroups in Banach spaces, Semigroup Forum38 (1989), 215–221.

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Hieber,Integrated semigroups and differential operators on L p, Dissertation, Tübingen, 1989.

  25. M. Hieber and H. Kellermann,Integrated semigroups, J. Funct. Anal.84 (1989), 160–180.

    Article  MATH  MathSciNet  Google Scholar 

  26. R.J. Hughes,Semigroups of unbounded linear operators in Banach space, Trans. Amer. Math. Soc.230 (1977), 113–145.

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Kantorovitz,The Hille-Yosida space of an arbitrary operator, J. Math. Anal. and Appl.136 (1988), 107–111.

    Article  MATH  MathSciNet  Google Scholar 

  28. T. Komura,Semigroups of operators in locally convex spaces, J. Funct. Anal.2 (1968), 258–296.

    Article  MATH  MathSciNet  Google Scholar 

  29. C. Martinez, M. Sanz and L. Marco,Fractional powers of operators, J. Math. Soc. Japan40 (1988), 331–347.

    MATH  MathSciNet  Google Scholar 

  30. C. Martinez, M. Sanz and L. Marco,Fractional powers of non-negative operators in Frechet spaces, Int. J. Math. and Math. Sci. 122 (1989), 309–320.

    Article  MATH  Google Scholar 

  31. C. Martinez and M. Sanz,n th-roots of a non-negative operator. Conditions for uniqueness, Manuscr. Math.64 (1989), 403–417.

    Article  MATH  MathSciNet  Google Scholar 

  32. I. Miyadera,On the generators of exponentially bounded C-semigroups, Proc. Japan Acad. Ser. A62 (1986), 239–242.

    Article  MATH  MathSciNet  Google Scholar 

  33. I. Miyadera,A generalization of the Hille-Yosida theorem, preprint (1988).

  34. I. Miyadera, S. Oharu and N. Okazawa,Generation theorems of semigroups of linear operators, Publ. R.I.M.S. Kyoto Univ.3 (1973), 509–555.

    Google Scholar 

  35. I. Miyadera and N. Tanaka,Exponentially bounded C-semigroups and generation of semigroups, preprint (1988).

  36. I. Miyadera and N. Tanaka,Exponentially bounded C-semigroups and integrated semigroups, Tokyo J. Math.12 (1989), 99–115.

    MATH  MathSciNet  Google Scholar 

  37. I. Miyadera and N. Tanaka,A remark on exponentially bounded C-semigroups, Proc. Japan Acad. Ser. A66 (1990), 31–35.

    MATH  MathSciNet  Google Scholar 

  38. R. Nagel (ed.),One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer-Verlag, Berlin, 1986.

    MATH  Google Scholar 

  39. F. Neubrander,Integrated semigroups and their application to the abstract Cauchy problem, Pac. J. Math.135 (1988), 111–155.

    MATH  MathSciNet  Google Scholar 

  40. F. Neubrander,Abstract elliptic operators, analytic interpolation semigroups, and Laplace transforms of analytic functions, Semesterbericht Funktionalanalysis, Tübingen, Wintersemester 1988/1989.

  41. F. Neubrander and B. Straub,Fractional powers of operators with polynomially bounded resolvent, Semesterbericht Funktionalanalysis, Tübingen, Wintersemester 1988/1989.

  42. A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.

    MATH  Google Scholar 

  43. B. Straub, Über generatoren von C-halbgruppen und Cauchyprobleme zweiter ordnung, Diplomarbeit, Tübingen (1989).

  44. N. Tanaka,On the exponentially bounded C-semigroups, Tokyo J. Math.10 (1987), 107–117.

    Article  MATH  MathSciNet  Google Scholar 

  45. N. Tanaka,Holomorphic C-semigroups and holomorphic semigroups, Semigroup Forum38 (1989), 253–263.

    Article  MATH  MathSciNet  Google Scholar 

  46. N. Tanaka,The convergence of exponentially bounded C-semigroups, preprint (1988).

  47. N. Tanaka,On perturbation theory for exponentially bounded C-semigroups, Semigroup Forum41 (1990), 215–237.

    Article  MATH  MathSciNet  Google Scholar 

  48. J. A. van Casteren,Generators of Strongly Continuous Semigroups, Research Notes in Math., 115, Pitman, London, 1985.

    MATH  Google Scholar 

  49. K. Yosida,Functional Analysis, Springer, Berlin, 1978.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio University, 45701, Athens, OH, USA

    Ralph deLaubenfels

Authors
  1. Ralph deLaubenfels
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

deLaubenfels, R. C-semigroups and strongly continuous semigroups. Israel J. Math. 81, 227–255 (1993). https://doi.org/10.1007/BF02761308

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761308

Keywords

Search

Navigation