Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators

  • 84 Accesses

  • 10 Citations


For any natural numberk, we characterize once-integrated Laplace transforms ofO((1+t)k) andO(t k) Banach-space-valued functions. We use this to give Hille-Yosida type characterizations of generators of polynomially bounded strongly continuous semigroups, related families of operators, and solutions of the abstract Cauchy problem.

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


  1. [1]

    W. Arendt,Vector-valued Laplace transforms and Cauchy problems Israel Journal of Mathematics59 (1987), 327–353.

  2. [2]

    M. Balabane, H. Emamirad and M. Jazar,Spectral distributions and generalization of Stone's theorem to the Banach space, Acta Applicandae Mathematicae31 (1993), 275–295.

  3. [3]

    J. M. Ball,Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proceedings of the American Mathematical Society63 (1977), 370–373.

  4. [4]

    R. deLaubenfels,Unbounded holomorphic functional calculus and abstract Cauchy problems for operators with polynomially bounded resolvens, Journal of Functional Analysis114 (1993), 348–394.

  5. [5]

    R. deLaubenfels,Existence families, functional calculi and evolution equations, Lecture Notes in Mathematics1570, Springer-Verlag, Berlin, 1994.

  6. [6]

    R. deLaubenfels and S. Kantorovitz,Laplace and Laplace-Stieltjes spaces, Journal of Functional Analysis,116 (1993), 1–61.

  7. [7]

    R. deLaubenfels, G. Sun and S. Wang,Regularized semigroups, existence families and the abstract Cauchy problem, Journal of Differential and Integral Equations8 (1995), 1477–1496.

  8. [8]

    H. Emamirad and M. Jazar,Applications of spectral distributions to some Cauchy problems in L p(R n), inSemigroup Theory and Evolution Equations: The Second International Conference, Delft 1989, Lecture Notes in Pure and Applied Mathematics, Vol. 135, Marcel Dekker, New York, 1991, pp. 143–151.

  9. [9]

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

  10. [10]

    B. Hennig, and F. Neubrander,On representations, inversions, and approximations of Laplace transforms in Banach spaces. Applicable Analysis49 (1993), 151–170.

  11. [11]

    M. Hieber, A. Holderrieth and F. Neubrander,Regularized semigroups and systems of linear partial differential equations, Annali della Scuola Normale Superiore di Pisa19 (1992), 363–379.

  12. [12]

    M. Hieber,Integrated semigroups and differential operators on L p spaces, Mathematische Annalen291 (1991), 1–16.

  13. [13]

    M. Jazar,Fractional powers of momentum of a spectral distribution, Proceedings of the American Mathematical Society, to appear.

  14. [14]

    S. Kantorovitz,The Hille-Yosida space of an arbitrary operator, Journal of Mathematical Analysis and Applications136 (1988), 107–111.

  15. [15]

    S. G. Krein, G. I. Laptev and G. A. Cretkova,On Hadamard correctness of the Cauchy problem for the equation of evolution. Soviet Mathematics Doklady11 (1970), 763–766.

  16. [16]

    Y. C. Li,Integrated C-semigroups and C-cosine functions of operators on locally convex spaces, Ph.D. dissertation, National Central University, 1991.

  17. [17]

    Y. C. Li and S. Y. Shaw,Integrated C-semigroups and the abstract Cauchy problem, preprint, 1993.

  18. [18]

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

  19. [19]

    J. A. van Casteren,Generators of Strongly Continuous Semigroups, Research Notes in Mathematics 115, Pitman, Boston, 1985.

  20. [20]

    J. M. A. M. van Neerven and B. Straub,On the existence and growth of mild solutions of the abstract Cauchy problem for operators with polynomially bounded resolvents, preprint, 1995.

  21. [21]

    S. Wang,Mild integrated C-existence families, Studia Mathematica112 (1995), 251–266.

  22. [22]

    D. V. Widder,An Introduction to Transform Theory, Academic Press, New York, 1971.

Download references

Author information

Correspondence to R. DeLaubenfels.

Additional information

Much of this work was done while the third author was visiting Ohio University. He would like to thank Ohio University, and Professors deLaubenfels and Swardson, for their hospitality and support.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

DeLaubenfels, R., Huang, Z., Wang, S. et al. Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators. Israel J. Math. 98, 189–207 (1997).

Download citation


  • Banach Space
  • Nonnegative Integer
  • Mild Solution
  • Closed Operator
  • Functional Calculus