Skip to main content
SpringerLink
Log in

A semigroup approach to fractional powers

  • Research article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We say that A has fractional powers {A t }t≥0 if there exists a nondegenerate C-regularized semigroup {W(t)}t≥0 such that A=C −1 W(1); then A t ≡C −1 W(t). We show that this generalizes the usual definition of fractional powers for nonnegative operators, and enables many operators with spectrum containing the entire unit disc to have fractional powers. Our definition gives clear, simple proofs of the basic properties of fractional powers. We show that, for nonnegative operators, the fractional powers with the property that, if A is of type θ, then A t is of type t θ, whenever t θ<π, are unique. More generally, for injective G∈B(X) commuting with A, we show that an operator A of G-regularized type θ has a unique family of fractional powers with the property that A t is of G-regularized type t θ whenever t θ<π. This leads to a construction of fractional powers of operators with polynomially bounded resolvent outside of an appropriate sector. We show that an operator is of regularized type if and only if it has exponentially bounded regularized imaginary powers.

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. Boyadzhiev, K.: Logarithms and imaginary powers of operators on Hilbert spaces. Collect. Math. 45, 287–300 (1994)

    MATH  MathSciNet  Google Scholar 

  2. Boyadzhiev, K., deLaubenfels, R.: Spectral theorem for unbounded strongly continuous groups on a Hilbert space. Proc. Am. Math. Soc. 120, 127–136 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cioranescu, I., Keyantuo, V.: Entire regularizations of strongly continuous groups and products of analytic generators. Proc. Am. Math. Soc. 128, 3587–3593 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Da Prato, G.: Semigruppi regolarizzabili. Ric. Mat. 15, 223–248 (1966)

    MATH  MathSciNet  Google Scholar 

  5. deLaubenfels, R.: Incomplete iterated Cauchy problems. J. Math. Anal. Appl. 168, 552–579 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. deLaubenfels, R.: Holomorphic C-existence families. Tokyo J. Math. 15, 17–38 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. deLaubenfels, R.: Existence Families, Functional Calculi and Evolution Equations. Lecture Notes in Mathematics, vol. 1570. Springer, New York (1994)

    MATH  Google Scholar 

  8. deLaubenfels, R.: Automatic extensions of functional calculi. Studia Math. 114, 237–259 (1995)

    MATH  MathSciNet  Google Scholar 

  9. deLaubenfels, R., Sun, G., Wang, S.: Regularized semigroups, existence families and the abstract Cauchy problem. J. Differ. Integral Equ. 8, 1477–1496 (1995)

    MATH  MathSciNet  Google Scholar 

  10. deLaubenfels, R., Yao, F., Wang, S.: Fractional powers and regularized type of an operator. J. Math. Anal. Appl. 199, 910–933 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196, 189–201 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fattorini, H.O.: The Cauchy Problem. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  13. Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford University Press, New York (1985)

    MATH  Google Scholar 

  14. Hughes, R.J.: Semigroups of unbounded linear operators in Banach spaces. Trans. Am. Math. Soc. 230, 113–145 (1977)

    Article  MATH  Google Scholar 

  15. Martinez, C., Sanz, M.: The Theory of Fractional Powers of Operators. North-Holland Mathematics Studies, vol. 187 (2001)

  16. Martinez, C., Sanz, M., Pastor, J.: A functional calculus and fractional powers for multivalued linear operators. Osaka J. Math. 37, 551–576 (2000)

    MATH  MathSciNet  Google Scholar 

  17. Monniaux, S.: A new approach to the Dore-Venni theorem. Math. Nachr. 204, 163–183 (1999)

    MATH  MathSciNet  Google Scholar 

  18. Nollau, V.: Über den Logartms abgeschlossener Operatoren in Banachschen Räumen. Acta Sci. Math. Szeged 30, 161–174 (1969)

    MATH  MathSciNet  Google Scholar 

  19. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    MATH  Google Scholar 

  20. Straub, B.: Fractional powers of operators with polynomially bounded resolvent and the semigroups generated by them. Hiroshima Math. J. 24, 529–548 (1994)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Ralph deLaubenfels

  2. Departament Matemàtica Aplicada, Universitat de València, 46100, Burjassot, València, Spain

    Javier Pastor

Authors
  1. Ralph deLaubenfels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Javier Pastor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ralph deLaubenfels.

Additional information

Communicated by Jerome A. Goldstein

This work was done while the second author was visiting Ohio University, with funding from Universitat de València. He would like to thank Ohio University and Professor deLaubenfels for their hospitality and support.

Rights and permissions

Reprints and permissions

About this article

Cite this article

deLaubenfels, R., Pastor, J. A semigroup approach to fractional powers. Semigroup Forum 76, 385–426 (2008). https://doi.org/10.1007/s00233-007-9023-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-007-9023-7

Keywords

Search

Navigation