On Products of Some Toeplitz Operators on Polyanalytic Fock Spaces

Abstract

The purpose of this paper is to study the Sarason’s problem on Fock spaces of polyanalytic functions. Namely, given two polyanalytic symbols f and g, we establish a necessary and sufficient condition for the boundedness of some Toeplitz products TfT subjected to certain restriction on f and g. We also characterize this property in terms of the Berezin transform.

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References

  1. [1]

    L. D. Abreu: On the structure of Gabor and super Gabor spaces. Monatsh. Math. 161 (2010), 237–253.

    MathSciNet  Article  Google Scholar 

  2. [2]

    A. Aleman, S. Pott, M. C. Reguera: Sarason conjecture on the Bergman space. Int. Math. Res. Not. 2017 (2017), 4320–4349.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    N. Askour, A. Intissar, Z. Mouayn: Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants. C. R. Acad. Sci., Paris, Sér. I, Math. 325 (1997), 707–712. (In French.)

    MathSciNet  Article  Google Scholar 

  4. [4]

    M. B. Balk: Polyanalytic Functions. Mathematical Research 63. Akademie Verlag, Berlin, 1991.

    Google Scholar 

  5. [5]

    H. Bommier-Hato, E. H. Youssfi, K. Zhu: Sarason’s Toeplitz product problem for a class of Fock spaces. Bull. Sci. Math. 141 (2017), 408–442.

    MathSciNet  Article  Google Scholar 

  6. [6]

    H. R. Cho, J.-D. Park, K. Zhu: Products of Toeplitz operators on the Fock space. Proc. Am. Math. Soc. 142 (2014), 2483–2489.

    MathSciNet  Article  Google Scholar 

  7. [7]

    A. Haimi, H. Hedenmalm: The polyanalytic Ginibre ensembles. J. Stat. Phys. 153 (2013), 10–47.

    MathSciNet  Article  Google Scholar 

  8. [8]

    F. Nazarov: A counterexample to Sarason’s conjecture. Available at http://users.math.msu.edu/users/fedja/prepr.html.

  9. [9]

    D. Sarason: Exposed points in H1. I. The Gohberg Anniversary Collection, Vol. II. Operator Theory: Advances and Applications 41. Birkhäuser, Basel, 1989, pp. 485–496.

    Google Scholar 

  10. [10]

    D. Sarason: Products of Toeplitz operators. Linear and Complex Analysis. Problem Book 3, Part I (V. P. Havin, N. K. Nikolskii, eds.). Lecture Notes in Mathematics 1573. Springer, Berlin, 1994, pp. 318–319.

    Google Scholar 

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Acknowledgements

This work is part of my PhD thesis. I am grateful to my advisers Professors S. Rigat and E. H. Youssfi for suggesting this problem and for helpful discussions.

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Correspondence to Irène Casseli.

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Casseli, I. On Products of Some Toeplitz Operators on Polyanalytic Fock Spaces. Czech Math J 70, 369–379 (2020). https://doi.org/10.21136/CMJ.2019.0334-18

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Keywords

  • polyanalytic function
  • Toeplitz operator
  • Fock space
  • Sarason’s problem

MSC 2010

  • 47B35
  • 30H20
  • 30G30
  • 46E22