The Range of the Berezin Transform

We give a description of all functions of the form \( \sum \limits_{i=1}^N{f}_i{\overline{g}}_i \) in the range of the Berezin transform, where f i and g i , 1 ≤ i ≤ N, are holomorphic functions on the unit disk 𝔻, which generalizes the corresponding result of Ahern. We also give a very simple proof of the results of Ahern and the results due to Čučković and Li who deal with the special case N = 2 and g 2 = z n.

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References

  1. 1.

    P. Ahern, “On the range of the Berezin transform,” J. Funct. Anal. 215, No. 1, 206–216 (2004).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    P. Ahern and Ž. Čučković, “A theorem of Brown–Halmos type for Bergman space Toeplitz operators,” J. Funct. Anal. 187, No. 1, 200–210 (2001).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ž. Čučković, “Berezin versus Mellin,” J. Math. Anal. and Appl. 287, No. 1, 234–243 (2003).

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    I. Louhichi, N. V. Rao, and A. Yousef, “Two questions on products of Toeplitz operators on the Bergman space,” Complex Anal. Oper. Theory 3, No. 4, 881–889 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Ž. Čučković and Bo Li, “Berezin transform, Mellin transform, and Toeplitz operators,” Complex Anal. Oper. Theory 6, 189-218 (2012).

  6. 6.

    D. Luecking, “Finite rank Toeplitz operators on the Bergman space,” Proc. Am. Math. Soc. 136, No. 5, 1717–1723 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    A. Alexandrov and G. Rozenblum, “Finite rank Toeplitz operators: Some extensions of D. Luecking’s theorem,” J. Funct. Anal. 256, No. 7, 2291–2303 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    K. Zhu, Operator Theory in Function Spaces, Am. Math. Soc., Providence, RI (2007).

  9. 9.

    P. Ahern, M. Flores, and W. Rudin, “An invariant volume-mean-value property,” J. Funct. Anal. 111, No. 2, 380–397 (1993).

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to N. V. Rao.

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Translated from Problemy Matematicheskogo Analiza 90, January 2018, pp. 73-81.

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Rao, N.V. The Range of the Berezin Transform. J Math Sci 228, 684–694 (2018). https://doi.org/10.1007/s10958-017-3656-1

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