Skip to main content
SpringerLink
Log in

Product formulas in functional calculi for sectorial operators

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study the product formula \((fg)(A) = f(A)g(A)\) in the framework of (unbounded) functional calculus of sectorial operators \(A\). We give an abstract result, and, as corollaries, we obtain new product formulas for the holomorphic functional calculus, an extended Stieltjes functional calculus and an extended Hille–Phillips functional calculus. Our results generalise previous work of Hirsch, Martinez and Sanz, and Schilling.

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. Batty, C.J.K., Chill, R., Tomilov, Yu.: Fine scales of decay of operator semigroups. J. Eur. Math. Soc. arXiv:1305.5365 (to appear)

  2. Clark, S.: Sums of operator logarithms. Q. J. Math. 60, 413–427 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  4. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, New York (2000)

  5. Gomilko, A., Haase, M., Tomilov, Yu.: Bernstein functions and rates in mean ergodic theorems for operator semigroups. J. Anal. Math. 118, 545–576 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Haase, M.: A general framework for holomorphic functional calculi. Proc. Edinb. Math. Soc. (2) 48, 423–444 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications 169. Birkhäuser, Basel (2006)

  8. Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups, 3rd printing of rev. ed. of 1957, Colloq. Publ. 31. AMS, Providence, RI (1974)

  9. Hirsch, F.: Intégrales de résolvantes et calcul symbolique. Ann. Inst. Fourier (Grenoble) 22, 239–264 (1972)

    Article  MATH  Google Scholar 

  10. Hirsch, F.: Domaines d’opérateurs représentés comme intègrales de résolvantes. J. Funct. Anal. 23, 199–217 (1976)

    Article  MATH  Google Scholar 

  11. Hirschman, I.I., Widder, D.V.: The Convolution Transform. Princeton University Press, Princeton (1955)

    Google Scholar 

  12. Karp, D., Prilepkina, E.: Generalized Stieltjes transforms: basic aspects. arXiv:1111.4271

  13. Kunstmann, P.C., Weis, L.: Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Math. vol. 1855. Springer, Berlin, pp. 65–311 (2004)

  14. Lancien, F., Lancien, G., Le Merdy, C.: A joint functional calculus for sectorial operators with commuting resolvents. Proc. Lond. Math. Soc. 77, 387–414 (1998)

    Article  Google Scholar 

  15. Martinez, C., Sanz, M.: An extension of the Hirsch symbolic calculus. Potential Anal. 9, 301–319 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Martinez, C., Sanz, M.: The Theory of Fractional Powers of Operators. North-Holland, Amsterdam (2001)

    MATH  Google Scholar 

  17. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and series. Elementary functions (Russian). Nauka, Moscow (1981)

  18. Schilling, R.L.: Subordination in the sense of Bochner and a related functional calculus. J. Aust. Math. Soc. Ser. A 64, 368–396 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions. de Gruyter Studies in Mathematics 37. Walter de Gruyter, Berlin (2010)

Download references

Author information

Authors and Affiliations

  1. St. John’s College, University of Oxford, Oxford, OX1 3JP, UK

    Charles Batty

  2. Faculty of Mathematics and Computer Science, Nicolas Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland

    Alexander Gomilko & Yuri Tomilov

  3. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956, Warsaw, Poland

    Yuri Tomilov

Authors
  1. Charles Batty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Gomilko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuri Tomilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles Batty.

Additional information

The research described in this paper was supported by the EPSRC Grant EP/J010723/1. The second and third authors were also partially supported by the NCN grant DEC-2011/03/B/ST1/00407 and by the EU Marie Curie IRSES program, Project “AOS”, No. 318910.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Batty, C., Gomilko, A. & Tomilov, Y. Product formulas in functional calculi for sectorial operators. Math. Z. 279, 479–507 (2015). https://doi.org/10.1007/s00209-014-1378-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-014-1378-3

Keywords

Mathematics Subject Classification

Search

Navigation