Skip to main content
SpringerLink
Log in

On spaces of vector-valued functions modeled on the Hardy and Bergman algebras

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The strong \(N^p(X), \mathcal {N}^p(X)\) and weak \(wN^p(X), w\mathcal {N}^p(X)\) spaces of analytic functions with values in a Banach space X, modeled on the large Hardy \(N^p\) and Bergman \(\mathcal {N}^p\) algebras respectively, are studied. The Fréchet envelopes and the continuous linear functionals on these spaces are described. We show that, although that strong and weak spaces are different, they have the same locally convex structure (the Fréchet envelopes are equal).

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. Blasco, O.: Boundary values of vector-valued harmonic functions considered as operators. Studia Math. 86, 19–33 (1987)

    Article  MathSciNet  Google Scholar 

  2. Blasco, O.: Hardy spaces of vector-valued functions: duality. Trans. Amer. Math. Sot. 308, 495–507 (1988)

    Article  MathSciNet  Google Scholar 

  3. Blasco, O.: Introduction to vector-valued Bergman spaces, in Function Spaces and Operator Theory (Joensuu, 2003), 9–30, Univ. Joensuu Dept. Math. Rep. Ser. 8, 9–30 (2005)

  4. Blasco, O., Drewnowski, L.: Extension of Pettis integration: Pettis operators and their integrals. Collectanea Math. 70(1), 267–281 (2019)

    Article  MathSciNet  Google Scholar 

  5. Bukhvalov, A.V.: Hardy spaces of vector-valued functions (in Russian). Zap. Nauchn. Sem. LOMI 65, 5–16 (1976)

    MATH  Google Scholar 

  6. Bukhvalov, A.V., Danilevich, A.A.: Boundary properties of analytic and harmonic functions with values in Banach spaces. Mat. Zametki 31, 203–214 (1982). (English translation: Math.Notes 31, 104–110 (1982))

    MathSciNet  MATH  Google Scholar 

  7. Bonet, J., Domański, P., Lindström, M.: Weakly compact composition operators on analytic vector-valued function spaces. Ann. Acad. Sci. Fenn. Math. 26, 233–248 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Duren, P.L.: Theory of \(H^p\) Spaces. Academic Press, New York and London (1979)

    MATH  Google Scholar 

  9. Drewnowski, L.: Topological vector groups and the Nevanlinna class. Funct. Approx. Comment. Math. 22, 25–39 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Duren, P.L., Romberg, B.W., Shields, A.L.: Linear Functionals on \(H^p\) with \(0<p<1\). J. Reine Angew. Math. 32–60, 238 (1969)

    MATH  Google Scholar 

  11. Eoff, C.: Fréchet envelops of certain algebras of analytic functions. Mich. Math. J. 35, 413–426 (1988)

    Article  MathSciNet  Google Scholar 

  12. Haldimann, A., Jarchow, H.: Nevanlinna algebras. Studia Math. 147(3), 243–268 (2001)

    Article  MathSciNet  Google Scholar 

  13. Kalton, N.J., Peck, N.T., Roberts, J.W.: F-Spaces Sampler, London Mathematical Society Lecture Note Series Book, vol. 89, Cambridge University Press, Cambridge (1984)

  14. Kim, H.O.: On an F-algebra of holomorphic functions. Can. J. Math. 40(3), 718–741 (1988)

    Article  MathSciNet  Google Scholar 

  15. Jarchow, H., Montesinos, V., Wirths, K.J., Xiao, J.: Duality for some Laree spaces of analytic functions. Proc. Edinb. Math. Soc. 44, 571–583 (2001)

    Article  MathSciNet  Google Scholar 

  16. Laitila, J., Tylli, H.-O., Wang, M.: Composition operators from weak to strong spaces of vector-valued analytic functions. J. Oper. Theory 62, 281–295 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Laitila, J., Tylli, H.-O.: Composition operators on vector-valued harmonic functions and Cauchy transforms. Indiana Univ. Math. J. 55(2), 719–746 (2006)

    Article  MathSciNet  Google Scholar 

  18. Nawrocki, M.: The Fréchet envelopes of vector-valued Smirnov classes. Stud. Math. 94, 61–75 (1989)

    Article  MathSciNet  Google Scholar 

  19. Nawrocki, M.: Duals of vector-valued \(H^p\)-spaces for \(0 < p < 1\). Indag. Math. N.S. 2(2), 233–241 (1991)

    Article  MathSciNet  Google Scholar 

  20. Nawrocki, M.: Köthe Sequence space representations of some weighted algebras of holomorphic functions. In: Functional Analysis, Proceedings of the First International Workshop held at Trier Univesity, Germany, W. De Gruyter, Berlin–New York, (1996)

  21. Nawrocki, M.: On the Smirnov class defined by the maximal function. CMB 45(2), 265–271 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Rolewicz, S.: Metric Linear Spaces. PWN-Polish Scientific Publishers, Warszawa; D. Reidel Publishing Company, Dordrecht, Boston, Lancaster (1984)

    MATH  Google Scholar 

  23. Shapiro, J.H.: Mackey topologies, reproducing Kernels, and diagonal maps on Hardy and Bergman spaces. Duke Math. J. 43, 187–202 (1976)

    Article  MathSciNet  Google Scholar 

  24. Shapiro, J.H., Shields, A.L.: Unusual topological properties of the Nevanlinna class. Amer. J. Math. 97, 915–936 (1976)

    Article  MathSciNet  Google Scholar 

  25. Stoll, M.: Mean Growth and Taylor coefficients of some topological algebras of holomorphic functions. Ann. Polon. Math. 35, 141–158 (1977)

    Article  Google Scholar 

  26. Stoll, M.: Mean Growth and Fourier coefficients of some classes of holomorphic functions on bounded symmetric domains. Ann. Polon. Math. 45, 161–183 (1985)

    Article  MathSciNet  Google Scholar 

  27. Yanagihara, N.: The containing Fréchet space for the class \(N^+\). Duke Math. J. 40, 93–103 (1973)

    Article  MathSciNet  Google Scholar 

  28. Yanagihara, N.: Multipliers and linear functionals for the class \(N^+\). Trans. Amer. Math. Soc. 180, 449–461 (1973)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to L. Drewnowski for turning his attention to the weak vector-valued Hardy spaces.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, A. Mickiewicz University in Poznań, Uniwersytetu Poznańskiego 4, 61–614, Poznan, Poland

    Marek Nawrocki

Authors
  1. Marek Nawrocki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marek Nawrocki.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nawrocki, M. On spaces of vector-valued functions modeled on the Hardy and Bergman algebras. RACSAM 114, 36 (2020). https://doi.org/10.1007/s13398-019-00773-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-019-00773-7

Keywords

Mathematics Subject Classification

Search

Navigation