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Integral Equations and Operator Theory

, Volume 44, Issue 1, pp 10–37 | Cite as

Toeplitz operators on the Fock space: Radial component effects

  • S. M. Grudsky
  • N. L. Vasilevski
Article

Abstract

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT a such that spT a =ess-spT a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.

Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.

AMS Classification

47B35 47E20 

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References

  1. [1]
    S. Axler and D. Zheng. Compact operators via the Berezin transform.Indiana Univ. Math. J. 47(2):387–400, 1998.Google Scholar
  2. [2]
    V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform.Comm. Pure Appl. Math., 3:187–214, 1961.Google Scholar
  3. [3]
    Harry Bateman and Arthur Erdélyi.Higher transcendental functions, Vol. 2. McGraw-Hill, 1954.Google Scholar
  4. [4]
    F. A. Berezin. Covariant and contravariant symbols of operators.Math. USSR Izvestia, 6:1117–1151, 1972.Google Scholar
  5. [5]
    F. A. Berezin and M. A. Shubin,The Schroedinger equation. Kluwer Academic Publishers, Dordrecht, Boston, 1991.Google Scholar
  6. [6]
    C. A. Berger and L. A. Coburn. Toeplitz operators and quantum mechanics.J. Funct. Analysis, 68:273–299, 1986.Google Scholar
  7. [7]
    C. A. Berger and L. A. Coburn. Toeplitz operators on the Segal-Bargmann space.Trans. of AMS, 301(2):813–829, 1987.Google Scholar
  8. [8]
    C. A. Berger and L. A. Coburn. Heat Flow and Berezin-Toeplitz estimates.Amer. J. Math., 116(3):563–590, 1994.Google Scholar
  9. [9]
    V. A. Fock. Konfigurationsraum und zweite Quantelung.Z. Phys., 75:622–647, 1932.Google Scholar
  10. [10]
    L. B. Folland,Harmonic Analysis in Phase Space. Princeton University Press, Princeton, New Jersey, 1989.Google Scholar
  11. [11]
    F. D. Gakhov.Boundary value problems. Int. Series of Monographs in Pure and Applied Mathematics. Vol. 85, Pergamon Press, 1966.Google Scholar
  12. [12]
    F. D. Gakhov and Yu. I. Cherskij.Equations of convolution type. (Uravneniya tipa svertki). Moskva: “Nauka”, 1978.Google Scholar
  13. [13]
    S. Grudsky and N. Vasilevski. Bergman-Toeplitz operators: Radial component influence. Integral Equations and Operator Theory, 40(1):16–33, 2001.Google Scholar
  14. [14]
    V. Guillemin. Toeplitz operators inn-dimensions.Integral Equations Operator Theory, 7:145–205, 1984.Google Scholar
  15. [15]
    E. Ramírez de Arellano and N. L. Vasilevski. Toeplitz operators on the Fock space with presymbols discontinuous on a thick set.Math. Nachr. 180:299–315, 1996.Google Scholar
  16. [16]
    E. Ramírez de Arellano and N. L. Vasilevski. Bargmann projection, three-valued functions and correspondind Toeplitz operators.Contemp. Math, 212:185–196, 1998.Google Scholar
  17. [17]
    Sangadji and K. Stroethoff. Compact Toeplitz operators on generalized Fock spaces.Acta Sci. Math. (Szeged), 64:657–669, 1998.Google Scholar
  18. [18]
    I. E. Segal.Lectures at the Summer Seminar on Appl. Math. Boulder, Colorado, 1960.Google Scholar
  19. [19]
    K. Stroethoff. Hankel and Toeplitz operators on the Fock space.Mich. Math. J., 39(1):3–16, 1992.Google Scholar
  20. [20]
    Sundaram Thangavelu.Lectures on Hermitte and Laguerre expansions. Princeton University Press, Preiceton, New Jersey, 1993.Google Scholar
  21. [21]
    N. Vasilevski, V. Kisil, E. Ramirez, and R. Trujillo. Toeplitz operators with discontinuous presymbols in the Fock space.Dokl. Math., 52(3):345–347, 1995.Google Scholar

Copyright information

© Birkhäuser Verlag 2002

Authors and Affiliations

  • S. M. Grudsky
    • 1
  • N. L. Vasilevski
    • 2
  1. 1.Department of MathematicsRostov-on-Don State UniversityRostov-on-DonRussia
  2. 2.Departmento de MatemáticasCINVESTAV del I.P.N.MéxicoMexico

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