Skip to main content
Log in

Application of the Laplace Transform of Tempered Distributions to the Construction of Functional Calculus

  • Published:
Ukrainian Mathematical Journal Aims and scope

An Erratum to this article was published on 01 May 2016

We use the generalized n-dimensional Laplace transform of tempered distributions whose supports are located in a positive n-dimensional cone to construct functional calculus for the commutative collections of injective generators of n-parameter analytic semigroups of operators acting in a Banach space.

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

Access this article

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, Ch. J. K. Batty, M. Hieber, and F. Neubrander, “Vector-valued Laplace Transforms and Cauchy Problems,” in: Monographs in Mathematics, 96, Birkhäser Verlag, Berlin (2011).

  2. C. Berg, K. Boyadzhiev, and R. Delaubenfels, “Generation of generators of holomorphic semigroups,” J. Australian Math. Soc., 55, No. 2, 246–269 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  3. P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Springer-Verlag, Berlin (1967).

    Book  MATH  Google Scholar 

  4. R. Delaubenfels, “Inverses of generators,” Proc. Amer. Math. Soc., 104, 443–448 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. A. Dubinskij, Sobolev Spaces of Infinite Order and Differential Equations, Kluwer, Dordrecht (1986).

    MATH  Google Scholar 

  6. V. I. Gorbachuk and A.V. Knyazyuk, “Boundary values of the solutions of operator-differential equations,” Rus. Math. Surv., 44, No. 3, 67–111 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Berlin (2006).

  8. E. Hille and R. Phillips, Functional Analysis and Semi-Groups, Vol. XXXI, AMS Coll. Publ., New York (1957).

  9. O. V. Lopushansky and S. V. Sharyn, “Generalized functional calculus of the Hille–Phillips type for multiparameter semigroups,” Sib. Math. J., 55, No. 1, 105–117 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. O. Lopushansky and S. Sharyn, “Operators commuting with multi-parameter shift semigroups,” Carpathian J. Math., 30, No. 2, 217–224 (2014).

    MathSciNet  MATH  Google Scholar 

  11. A. R. Mirotin, “On some properties of the multidimensional Bochner–Phillips functional calculus,” Sib. Math. J., 52, No. 6, 1032–1041 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  12. E. A. Nelson, “Functional calculus using singular Laplace integrals,” Trans. Amer. Math. Soc., 88, 400–413 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  13. R. S. Phillips, “Spectral Theory for Semigroups of Linear Operators,” Trans. Amer. Math. Soc., 71, 393–415 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Schaefer, Topological Vector Spaces, Springer-Verlag, Berlin (1971).

    Book  MATH  Google Scholar 

  15. L. Schwartz, “Espaces de fonctions différentielles à valeurs vectorielles,” J. An. Math., 4, 88–148 (1954/55).

  16. R. T. Seeley, “Extensions of C -functions defined in a half space,” Proc. Amer. Math. Soc., 15, 625–626 (1964).

    MathSciNet  MATH  Google Scholar 

  17. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York (1975).

  18. P. Richter, Unitary Representations of Countable Inifinite-Dimensional Lie Groups, MPH 5, Leipzig Univesität (1977).

  19. V. S. Vladimirov, Generalized Functions in Mathematical Physics, Mir, Moscow (1979).

  20. I. I. Vrabie, C 0 -Semigroup and Applications, Elsevier, New York–Amsterdam (2003).

  21. V. V. Zharinov, “Compact families of locally convex topological vector spaces. Fréchet–Schwartz and dual Fréchet–Schwartz spaces,” Rus. Math. Surv., 34, No. 4, 105–143 (1979).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 11, pp. 1498–1511, November, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lopushans’kyi, O.V., Sharyn, S.V. Application of the Laplace Transform of Tempered Distributions to the Construction of Functional Calculus. Ukr Math J 67, 1687–1703 (2016). https://doi.org/10.1007/s11253-016-1183-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-016-1183-8

Navigation