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Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

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Abstract

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for S φ S ψ = S φψ .

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References

  1. Yu, T., Wu, S.: Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space. Acta Math. Sinica, English Series, to appear

  2. Stroethoff, K., Zheng, D.: Algebraic and spectral properties of dual Toeplitz operaors. Trans. Amer. Math. Soc., 354, 2495–2520 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Lu, Y.: Commuting dual Toeplitz operators with pluriharmonic symbols. J. Math. Anal. Appl., 302, 149–156 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lu, Y., Shang, S.: Commuting dual Toeplitz operators on the polydisc. Acta Mathematica Sinica, English Series, 23(5), 857–868 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheng, G., Yu, T.: Commuting dual Toeplitz operators on the Bergman space of the polydisc. Chin. Quart. J. Math., to appear

  6. Yu, T., Cheng, G.: Essentially commuting dual Toeplitz operators on the polydisc. Acta Mathematica Sinica Chinese Series, 50(5), 1007–1016 (2007)

    MATH  MathSciNet  Google Scholar 

  7. Wu, Z.: Hankel and Toeplitz operators on Dirichlet spaces. Integ. Equ. Oper. Theory, 15, 503–525 (1992)

    Article  MATH  Google Scholar 

  8. Wu, Z.: Operator theory and function theory on Dirichlet space, “Analytic Spaces” ed. by S. Axler, J. McCarthy and D. Sarason, MSRI, 179–199, 1998

  9. Cao, G.: Fredholm properties of Toeplitz operators on Dirichlet space. Pacific J. Math., 188(2), 209–224 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Duistermaat, J. J., Lee, Y. J.: Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl., 300, 54–67 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lee, Y. J.: Algebraic properties of Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl., 329(2), 1361–1329 (2007)

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  13. Axler, S., Čučkovič, Z.: Commuting Toeplitz operators with harmonic symbols. Integ. Equa. Oper. Theory, 14, 1–12 (1991)

    Article  MATH  Google Scholar 

  14. Axler, S., Čučkovič, Ž., Rao, N. V.: Commutants of analytic Toeplitz Operators on the Bergman space. Proc. Amer. Math. Soc., 128, 1951–1953 (1999)

    Article  Google Scholar 

  15. Choe, B. R., Lee, Y. J.: Commutants of analytic Toeplitz Operators on the harmonic Bergman space. Integr. Equ. Oper. Theory, 50, 559–564 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, P. R. China

    Tao Yu & Shi Yue Wu

  2. Department of Mathematics, Jilin University, Changchun, 130024, P. R. China

    Tao Yu

Authors
  1. Tao Yu
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  2. Shi Yue Wu
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Correspondence to Tao Yu.

Additional information

The first author is partly supported by the NNSFC (No. 10771064)

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Yu, T., Wu, S.Y. Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space. Acta. Math. Sin.-English Ser. 24, 1843–1852 (2008). https://doi.org/10.1007/s10114-008-7266-1

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  • DOI: https://doi.org/10.1007/s10114-008-7266-1

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