Browse > Article
http://dx.doi.org/10.4134/CKMS.c180148

COMMUTANTS OF TOEPLITZ OPERATORS WITH POLYNOMIAL SYMBOLS ON THE DIRICHLET SPACE  

Chen, Yong (Department of Mathematics Hangzhou Normal University)
Lee, Young Joo (Department of Mathematics Chonnam National University)
Publication Information
Communications of the Korean Mathematical Society / v.34, no.2, 2019 , pp. 533-542 More about this Journal
Abstract
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.
Keywords
commutant; Toeplitz operator; Dirichlet space;
Citations & Related Records
연도 인용수 순위
  • Reference
1 S. Axler and Z. Cuckovic, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory 14 (1991), no. 1, 1-12.   DOI
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.   DOI
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.   DOI
5 C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1-31.   DOI
6 Z. Cuckovic, Commutants of Toeplitz operators on the Bergman space, Pacific J. Math. 162 (1994), no. 2, 277-285.   DOI
7 J. A. Deddens and T. K. Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973), 261-273.   DOI
8 D. A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113-139.   DOI
9 J. J. Duistermaat and Y. J. Lee, Toeplitz operators on the Dirichlet space, J. Math. Anal. Appl. 300 (2004), no. 1, 54-67.   DOI
10 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.
11 A. Tikaradze, Centralizers of Toeplitz operators with polynomial symbols, Complex Anal. Oper. Theory 6 (2012), no. 6, 1275-1279.   DOI
12 K. Zhu, Operator Theory in Function Spaces, second edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007.
13 J. E. Thomson, The commutants of certain analytic Toeplitz operators, Proc. Amer. Math. Soc. 54 (1976), 165-169.   DOI
14 Z. Wu, Carleson measures and multipliers for Dirichlet spaces, J. Funct. Anal. 169 (1999), no. 1, 148-163.   DOI
15 K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, 226, Springer-Verlag, New York, 2005.