Fock Spaces

  • Kehe Zhu
Part of the Graduate Texts in Mathematics book series (GTM, volume 263)


In this chapter, we define Fock spaces and prove basic properties about them. The following topics are covered in this chapter: reproducing kernel, integral representation, duality, complex interpolation, atomic decomposition, translation invariance, and a version of the maximum modulus principle.


Integral Operator Entire Function Composition Operator Heisenberg Group Closed Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 46.
    B. Carswell, B. MacCluer, A. Schuster, Composition operators on the Fock space. Acta. Sci. Math. (Szeged) 69, 871–887 (2003)MathSciNetMATHGoogle Scholar
  2. 74.
    M. Dostanić, K. Zhu, Integral operators induced by the Fock kernel. Integr. Equat. Operat. Theor. 60, 217–236 (2008)MATHCrossRefGoogle Scholar
  3. 76.
    P. Duren, Theory of H p Spaces, 2nd edn, (Dover Publications, New York, 2000)Google Scholar
  4. 96.
    O. Furdui, On a class of integral operators. Integr. Equat. Operat. Theor. 60, 469–483 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 110.
    K. Guo, K. Izuchi, Composition operators on Fock type spaces. Acta Sci. Math. (Szeged) 74, 807–828 (2008)MathSciNetMATHGoogle Scholar
  6. 113.
    P. Halmos, V. Sunder, Bounded Integral Operator on L 2 Spaces, (Springer, Berlin, 1978)Google Scholar
  7. 117.
    W. Hayman, On a conjecture of korenblum. Anal. (Munich) 19, 195–205 (1999)MathSciNetMATHGoogle Scholar
  8. 119.
    H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, (Springer, New York, 2000)MATHCrossRefGoogle Scholar
  9. 122.
    A. Hinkkanen, On a maximum principle in Bergman space. J. Anal. Math. 79, 335–344 (1999)MathSciNetMATHCrossRefGoogle Scholar
  10. 138.
    S. Janson, J. Peetre, R. Rochberg, Hankel forms and the Fock space. Revista Mat. Ibero-Amer. 3, 61–138 (1987)MathSciNetMATHCrossRefGoogle Scholar
  11. 141.
    B. Korenblum, A maximum principle for the Bergman space. Publ. Mat. 35, 479–486 (1991)MathSciNetMATHGoogle Scholar
  12. 142.
    S. Krantz, Function Theory of Several Complex Variables, 2nd edn. (American Mathematical Society, Providence, RI, 2001)MATHGoogle Scholar
  13. 177.
    A.M. Perelomov, Generalized Coherent States and Their Applications, (Springer, Berlin, 1986)MATHCrossRefGoogle Scholar
  14. 194.
    A. Schuster, The maximum principle for the Bergman space and the Möbius pseudodistance for the annulus. Proc. Amer. Math. Soc. 134, 3525–3530 (2006)MathSciNetMATHCrossRefGoogle Scholar
  15. 214.
    P. Sjögren, Un contre-exemple pour le noyau reproduisant de la mesure gaussienne dans le plan complexe, Seminaire Paul Krée (Equations aux dérivées partienlles en dimension infinite) 1975/76, ParisGoogle Scholar
  16. 215.
    E. Stein, Interpolation of linear operators. Trans. Amer. Math. Soc. 83, 482–492 (1956)MathSciNetMATHCrossRefGoogle Scholar
  17. 216.
    E. Stein, G. Weiss, Interpolation of operators with change of measures. Trans. Amer. Math. Soc. 87, 159–172 (1958)MathSciNetMATHCrossRefGoogle Scholar
  18. 217.
    R. Strichartz, L p contractive projections and the heat semigroup for differential forms. J. Funct. Anal. 65, 348–357 (1986)MathSciNetMATHCrossRefGoogle Scholar
  19. 224.
    J.Y. Tung, Taylor coefficients of functions in Fock spaces. J. Math. Anal. Appl. 318, 397–409 (2006)MathSciNetMATHCrossRefGoogle Scholar
  20. 226.
    J.Y. Tung, On Taylor coefficients and multipliers in Fock spaces. Contemp. Math. 454, 135–147 (2008)MathSciNetCrossRefGoogle Scholar
  21. 231.
    R. Wallsten, The S p-criterion for Hankel forms on the Fock space, 0 < p < 1. Math. Scand. 64, 123–132 (1989)MathSciNetMATHGoogle Scholar
  22. 232.
    C. Wang, Some results on Korenblum’s maximum principle J. Math. Anal. Appl. 373, 393–398 (2011)MathSciNetMATHCrossRefGoogle Scholar
  23. 250.
    K. Zhu, Operator Theory in Function Spaces, 2nd edn. (American Mathematical Society, Providence, RI, 2007)MATHGoogle Scholar
  24. 255.
    K. Zhu, Invariance of Fock spaces under the action of the Heisenberg group. Bull. Sci. Math. 135, 467–474 (2011)MathSciNetMATHCrossRefGoogle Scholar
  25. 256.
    K. Zhu, Duality of Bloch spaces and norm convergence of Taylor series. Michigan Math. J. 38, 89–101 (1991)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Kehe Zhu
    • 1
  1. 1.Department of Mathematics and StatisticsState University of New YorkAlbanyUSA

Personalised recommendations