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Two Approaches to Toeplitz Operators on Fock Space

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Quantization and Infinite-Dimensional Systems

Abstract

The concrete model of the Fock space as an L 2 -space of entire functions established by Segal10 and developed by Bargmann1 provides a convenient and mathematically precise language for studying free Bose fields. C. Berger and L. Coburn2,3 (using earlier ideas of F.A. Berezin and W. Arveson) have proposed to view a broad family of observables as Toeplitz operators. This allows to unify the operator-theoretic study, linking it with function theory via symbol analysis. Related results for special types of analytic symbols can be traced much earlier in a paper by Newman with Shapiro.9

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References

  1. V. Bargmann, On a Hilbert space of analytic functions and associated integral transform, Comm. Pure Appl. Math 14:187 (1961).

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  2. C.A. Berger and L.A. Coburn, Toeplitz operators and quantum mechanics, J. Funct. Anal. 68:273 (1986).

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  3. C.A. Berger and L.A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301:813 (1987).

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  4. J. Janas, Inductive limit of operators and its applications, Studia Math. 90:87 (1988).

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  5. J. Janas, Unbounded Toeplitz operators in the Bargmann-Segal space, Studia Math. 99:87 (1991).

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  6. J. Janas and K. Rudol, Toeplitz operators on the Segal-Bargmann space of infinitely many variables Operator Theory: Adv. Appl. 43:87 (1990).

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  7. J. Janas and J.Stochel, Unbounded Toeplitz operators on the Segal-Bargmann space II, J. Func tional Anal. ,to appear.

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  8. A.V. Marchenko, Selfadjoint differential operators with an infinite number of independent variables, Mat. Sbornik 96 (2):276 (1975) (in Russian).

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  9. D.J.Newman and H. Shapiro, Fisher spaces of entire functions, in: “Entire Functions and Related Parts of Analysis”,J. Korevaar, ed., Proc. Symp. Pure Math. 11 Amer. Math. Soc., Providence (1968).

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  10. I.E. Segal, The complex wave representation of the free boson field, Adv. in Math, Suppl. Studies 3:321 (1978).

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Authors and Affiliations

  1. Institute of Mathematics of the Polish Academy of Sciences, Ul.Sw.Tomasza 30, 31-027, Krakow, Poland

    J. Janas & K. Rudol

Authors
  1. J. Janas
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  2. K. Rudol
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Editor information

Editors and Affiliations

  1. Catholic University of Louvain, Louvain-la-Neuve, Belgium

    J-P. Antoine

  2. Concordia University, Montréal, Québec, Canada

    S. Twareque Ali

  3. Warsaw University, Białystok, Poland

    W. Lisiecki  & A. Odzijewicz  & 

  4. Bulgarian Academy of Sciences, Sofia, Bulgaria

    I. M. Mladenov

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© 1994 Springer Science+Business Media New York

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Janas, J., Rudol, K. (1994). Two Approaches to Toeplitz Operators on Fock Space. In: Antoine, JP., Ali, S.T., Lisiecki, W., Mladenov, I.M., Odzijewicz, A. (eds) Quantization and Infinite-Dimensional Systems. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2564-6_1

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  • DOI: https://doi.org/10.1007/978-1-4615-2564-6_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6095-7

  • Online ISBN: 978-1-4615-2564-6

  • eBook Packages: Springer Book Archive

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