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Carleson measures and Douglas’ question on the Bergman space

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Abstract

Motivated by Douglas’ question about the invertibility of Toeplitz operators on the Hardy space, we study a related question concerning the Berezin transform and averaging function of a Carleson measure for the weighted Bergman space of the disc. As a consequence, we obtain a necessary and sufficient condition for the invertibility of Toeplitz operators whose symbols are averaging functions of these Carleson measures.

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References

  1. Devinatz, A.: Toeplitz operators on \(H^{2}\) spaces. Trans. Am. Math. Soc. 112, 304–317 (1964)

    MATH  Google Scholar 

  2. Douglas, R.G.: Banach Algebra Techniques in the Theory of Toeplitz Operators. American Mathematical Society, Providence, RI (1973), Expository Lectures from the CBMS Regional Conference held at the University of Georgia, Athens, GA, June 12–16 (1972), Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 15

  3. Duren, P., Schuster, A.: Bergman Spaces, Mathematical Surveys and Monographs, vol. 100. American Mathematical Society, Providence, RI (2004)

    MATH  Google Scholar 

  4. Luecking, D.H.: Inequalities on Bergman spaces. Ill. J. Math. 25(1), 1–11 (1981)

    MathSciNet  MATH  Google Scholar 

  5. Luecking, D.H.: Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. Am. J. Math. 107(1), 85–111 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. McDonald, G., Sundberg, C.: Toeplitz operators on the disc. Indiana Univ. Math. J. 28(4), 595–611 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Oleinik, V.L., Pavlov, B.S.: Embedding theorems for the weighted class of harmonic and analytic functions. J. Soviet Math. 2, 135–142 (1974)

    Article  Google Scholar 

  8. Wolff, T.H.: Counterexamples to two variants of the Helson–Szegö theorem. J. Anal. Math. 88, 41–62 (2002). (Dedicated to the memory of Tom Wolff)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhao, X., Zheng, D.: Invertibility of Toeplitz operators via Berezin transforms. J. Oper. Theory 75(2), 475–495 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhu, K.: Operator theory in function spaces. In: Mathematical Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence, RI (2007)

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  1. Department of Mathematics, University of Toledo, Toledo, OH, 43606, USA

    Željko Čučković & Anthony Vasaturo

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  1. Željko Čučković
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  2. Anthony Vasaturo
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Correspondence to Anthony Vasaturo.

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Čučković, Ž., Vasaturo, A. Carleson measures and Douglas’ question on the Bergman space. Rend. Circ. Mat. Palermo, II. Ser 67, 323–336 (2018). https://doi.org/10.1007/s12215-017-0317-7

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  • DOI: https://doi.org/10.1007/s12215-017-0317-7

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