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Ranks of commutators and generalized semicommutators of quasihomogeneous Toeplitz operators

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Abstract

This paper is devoted to solve the finite rank problem for the commutator and generalized semicommutator of Toeplitz operators with quasihomogeneous symbols acting on both the harmonic Bergman space and the Bergman space. In particular, when one of quasihomogeneous symbols is the form of \(e^{ik\theta }r^{m}\), we first obtain specific sufficient and necessary conditions for the commutator and generalized semicommutator to be finite rank. Then we make further efforts to determine the range of the finite rank commutator and generalized semicommutator, and consequently the rank and the explicit canonical form are obtained. As applications, several interesting corollaries and nontrivial examples are given.

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References

  1. Ahern, P., Čučković, Ž.: A theorem of Brown–Halmos type for Bergman space Toeplitz operators. J. Funct. Anal. 187, 200–210 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Axler, S., Čučković, Ž.: Commuting Toeplitz operators with harmonic symbols. Integral Equ. Oper. Theory 14, 1–12 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, 4th printing, with Corrections. National Bureau of Standards, Washington (1965)

  4. Axler, S., Chang, S., Sarason, D.: Product of Toeplitz operators. Integral Equ. Oper. Theory 1, 285–309 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Bauer, W., Issa, H.: Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal–Bargmann space. J. Math. Anal. Appl. 386, 213–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bauer, W., Lee, Y.J.: Commuting Toeplitz operators on the Segal-Bargmann space. J. Funct. Anal. 260, 460–489 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Y., Koo, H., Lee, Y.J.: Ranks of commutators of Toeplitz operators on the harmonic Bergman space. Integral Equ. Oper. Theory 75, 31–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Choe, B.R., Lee, Y.J.: Commuting Toeplitz operators on the harmonic Bergman spaces. Mich. Math. J. 46, 163–174 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Čučković, Ž.: Finite rank perturbations of Toeplitz operators. Integral Equ. Oper. Theory 59, 345–353 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Čučković, Ž., Louhichi, I.: Finite rank commutators and semicommutators of quasihomogeneous Toeplitz operators. Complex Anal. Oper. Theory 2, 429–439 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Čučković, Ž., Rao, N.V.: Mellin transform, monomial symbols, and commuting Toeplitz operators. J. Funct. Anal. 154, 195–214 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ding, X., Zheng, D.: Finite rank commutator of Toeplitz operators or Hankel operators. Houst. J. Math. 34, 1099–1119 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Dong, X.T., Zhou, Z.H.: Algebraic properties of Toeplitz operators with separately quasihomogeneous symbols on the Bergman space of the unit ball. J. Oper. Theory. 66, 193–207 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Dong, X.T., Zhou, Z.H.: Products of Toeplitz operators on the harmonic Bergman space. Proc. Am. Math. Soc. 138, 1765–1773 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dong, X.T., Zhou, Z.H.: Commuting quasihomogeneous Toeplitz operators on the harmonic Bergman space. Complex Anal. Oper. Theory 7, 1267–1285 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dong, X.T., Zhou, Z.H.: Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space. Stud. Math. 219, 163–175 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dong, X.T., Liu, C., Zhou, Z.H.: Quasihomogeneous Toeplitz operators with integrable symbols on the harmonic Bergman space. Bull. Aust. Math. Soc. 90, 494–503 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guo, K., Sun, S., Zheng, D.: Finite rank commutators and semicommutators of Toeplitz operators with harmonic symbols. Ill. J. Math. 51, 583–596 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Louhichi, I., Strouse, E., Zakariasy, L.: Products of Toeplitz operators on the Bergman space. Integral Equ. Oper. Theory 54, 525–539 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Louhichi, I., Zakariasy, L.: On Toeplitz operators with quasihomogeneous symbols. Arch. Math. 85, 248–257 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Louhichi, I., Zakariasy, L.: Quasihomogeneous Toeplitz operators on the harmonic Bergman space. Arch. Math. 98, 49–60 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Quiroga-Barranco, R., Sanchez-nungaray, A.: Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of lagrangian frames. J. Oper. Theory 71, 199–222 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Quiroga-Barranco, R., Sanchez-nungaray, A.: Toeplitz operators with quasi-homogeneuos quasi-radial symbols on some weakly pseudoconvex domains. Complex Anal. Oper. Theory 9, 1111–1134 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Vasilevski, N.: Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators. Integral Equ. Oper. Theory 66, 141–152 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zheng, D.: Commuting Toeplitz operators with pluriharmonic symbols. Trans. Am. Math. Soc. 350, 1595–1618 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhou, Z.H., Dong, X.T.: Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball. Integral Equ. Oper. Theory 64, 137–154 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11201331; 11371276; 11401431).

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300354, People’s Republic of China

    Xing-Tang Dong & Ze-Hua Zhou

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  1. Xing-Tang Dong
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  2. Ze-Hua Zhou
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Correspondence to Xing-Tang Dong.

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Communicated by A. Constantin.

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Dong, XT., Zhou, ZH. Ranks of commutators and generalized semicommutators of quasihomogeneous Toeplitz operators. Monatsh Math 183, 103–141 (2017). https://doi.org/10.1007/s00605-017-1033-2

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  • DOI: https://doi.org/10.1007/s00605-017-1033-2

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