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Sums of Products of Toeplitz and Hankel Operators on the Dirichlet Space

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Abstract

On the Dirichlet space of the unit disk, we consider operators that are finite sums of Toeplitz products, Hankel products or products of a Toeplitz operator and a Hankel operator. We characterize when such operators are equal to zero. Our results extend several known results using completely different arguments.

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References

  1. Adams R.: Sobolev Spaces. Academic press, New York (1975)

    MATH  Google Scholar 

  2. Chen Y.: Commuting Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl. 357, 214–224 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen Y., Nguyen Q.D.: Toeplitz and Hankel operators with L ∞,1 symbols on the Dirichlet space. J. Math. Anal. Appl. 369, 368–376 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Choe B.R., Koo H.: Zero products of Toeplitz operators with harmonic symbols. J. Funct. Anal. 233, 307–334 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gu C., Zheng D.: Products of block Toeplitz operators. Pacific J. Math. 185, 115–148 (1998)

    Article  MathSciNet  Google Scholar 

  6. Lee Y.J.: Algebraic properties of Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl. 329, 1316–1329 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lee Y.J.: Finite sums of Toeplitz products on the Dirichlet space. J. Math. Anal. Appl. 357, 504–515 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lee Y.J., Nguyen Q.D.: Toeplitz operators on the Dirichlet spaces of planar domains. Proc. Am. Math. Soc. 139, 547–558 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wu, Z.: Operator theory and function theory on the Dirichlet space. In: Axler, S., McCarthy, J., Sarason D. (eds.) Holomorphic Spaces, pp. 179–199. MSRI (1998)

  10. Yu T.: Toeplitz operators on the Dirichlet space. Integral Equ. Oper. Theory 67, 163–170 (2010)

    Article  MATH  Google Scholar 

  11. Zhao L.: Hankel operators on the Dirichlet space. J. Math. Anal. Appl. 352, 767–772 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhu K.: Operator Theory in Function Spaces, 2nd edn. American Mathematical Society, Providence (2007)

    MATH  Google Scholar 

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Author notes

  1. Young Joo Lee

    Present address: Department of Mathematics and Statistics, SUNY at Albany, Albany, NY, 12222, USA

Authors and Affiliations

  1. Department of Mathematics, Chonnam National University, Gwangju, 500-757, Korea

    Young Joo Lee

  2. Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY, 12222, USA

    Kehe Zhu

Authors
  1. Young Joo Lee
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  2. Kehe Zhu
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Corresponding author

Correspondence to Kehe Zhu.

Additional information

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government(NRF-2010-013-C00005).

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Lee, Y.J., Zhu, K. Sums of Products of Toeplitz and Hankel Operators on the Dirichlet Space. Integr. Equ. Oper. Theory 71, 275–302 (2011). https://doi.org/10.1007/s00020-011-1901-4

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  • DOI: https://doi.org/10.1007/s00020-011-1901-4

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