• Chen, Yong (Department of Mathematics Hangzhou Normal University) ;
  • Lee, Young Joo (Department of Mathematics Chonnam National University)
  • Received : 2018.04.11
  • Accepted : 2018.05.16
  • Published : 2019.04.30


We study commutants of Toeplitz operators acting on the Dirichlet space of the unit disk and prove that an operator in the Toeplitz algebra commuting with a Toeplitz operator with a nonconstant polynomial symbol must be a Toeplitz operator with an analytic symbol.


Supported by : NSFC, ZJNSFC, National Research Foundation of Korea(NRF)


  1. S. Axler and Z. Cuckovic, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory 14 (1991), no. 1, 1-12.
  2. A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89-102.
  3. Y. Chen, Commuting Toeplitz operators on the Dirichlet space, J. Math. Anal. Appl. 357 (2009), no. 1, 214-224.
  4. Y. Chen and N. Q. Dieu, Toeplitz and Hankel operators with $L^{{\infty},1}$ symbols on Dirichlet space, J. Math. Anal. Appl. 369 (2010), no. 1, 368-376.
  5. C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1-31.
  6. Z. Cuckovic, Commutants of Toeplitz operators on the Bergman space, Pacific J. Math. 162 (1994), no. 2, 277-285.
  7. J. A. Deddens and T. K. Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973), 261-273.
  8. J. J. Duistermaat and Y. J. Lee, Toeplitz operators on the Dirichlet space, J. Math. Anal. Appl. 300 (2004), no. 1, 54-67.
  9. O. El-Fallah, K. Kellay, J. Mashreghi, and T. Ransford, A primer on the Dirichlet space, Cambridge Tracts in Mathematics, 203, Cambridge University Press, Cambridge, 2014.
  10. D. A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113-139.
  11. A. Tikaradze, Centralizers of Toeplitz operators with polynomial symbols, Complex Anal. Oper. Theory 6 (2012), no. 6, 1275-1279.
  12. J. E. Thomson, The commutants of certain analytic Toeplitz operators, Proc. Amer. Math. Soc. 54 (1976), 165-169.
  13. Z. Wu, Carleson measures and multipliers for Dirichlet spaces, J. Funct. Anal. 169 (1999), no. 1, 148-163.
  14. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, 226, Springer-Verlag, New York, 2005.
  15. K. Zhu, Operator Theory in Function Spaces, second edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007.