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Toeplitz operators with BMO symbols on the weighted Bergman space of the unit ball

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Abstract

In this paper we prove that the boundedness and compactness of Toeplitz operator with a BMO 1 α symbol on the weighted Bergman space A 2 α (B n ) of the unit ball is completely determined by the behavior of its Berezin transform, where α > −1 and n ≥ 1.

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References

  1. Zhu, K.: Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Operator Theory, 20, 329–357 (1988)

    MathSciNet  MATH  Google Scholar 

  2. Békollé, D., Berger, C. A., Coburn, L. A., et al.: BMO in the Bergman metric on bounded symmetric domains. J. Funct. Anal., 93, 310–350 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Engliš, M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integr. Equ. Oper. Theory, 33, 426–455 (1999)

    Article  MATH  Google Scholar 

  4. Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2004

    Google Scholar 

  5. Axler, S., Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J., 47, 387–400 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zorboska, N.: Toeplitz operators with BMO symbols and the Berezin transform. Int. J. Math. Sci., 46, 2929–2945 (2003)

    Article  MathSciNet  Google Scholar 

  7. Zhu, K.: VMO, ESV and Toeplitz operators on the Bergman space. Trans. Amer. Math. Soc., 302, 617–646 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berger, C. A., Coburn, L. A., Zhu, K.: Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus. Amer. J. Math., 110, 921–953 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, H. P., Luecking, D. H.: BMO on strongly pseudoconvex domains: Hankel operators, duality and \(\bar \partial \)-estimates. Trans. Amer. Math. Soc., 346, 661–691 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhu, K.: BMO and Hankel operators on Bergman spaces. Pacific J. Math., 155, 377–395 (1992)

    MathSciNet  MATH  Google Scholar 

  11. Luecking, D. H.: Trace ideal criteria for Toeplitz operators. J. Funct. Anal., 73, 345–368 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yu, T., Wu, S. Y.: Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space. Acta Mathematica Sinica, English Series, 25, 245–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brown, A., Halmos, P.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math., 213, 89–102 (1964)

    MathSciNet  Google Scholar 

  14. Rudin, W.: Function Theory in the Unit Ball of Cn, Springer-Verlag, New York, 1980

    MATH  Google Scholar 

  15. Stroethoff, K., Zheng, D.: Bounded Toeplitz products on Bergman spaces of the unit ball. J. Math. Anal. Appl., 325, 114–129 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P. R. China

    Kan Zhang & Yu Feng Lu

  2. College of Sciences, Shenyang Agricultural University, Shenyang, 110161, P. R. China

    Kan Zhang

  3. School of Science, Dalian Jiaotong University, Dalian, 116028, P. R. China

    Chao Mei Liu

Authors
  1. Kan Zhang
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  2. Chao Mei Liu
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  3. Yu Feng Lu
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Correspondence to Yu Feng Lu.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 10671028, 10971020)

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Zhang, K., Liu, C.M. & Lu, Y.F. Toeplitz operators with BMO symbols on the weighted Bergman space of the unit ball. Acta. Math. Sin.-English Ser. 27, 2129–2142 (2011). https://doi.org/10.1007/s10114-011-0038-3

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  • DOI: https://doi.org/10.1007/s10114-011-0038-3

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