Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols

  • Nanxing Guan
  • Xianfeng ZhaoEmail author


Let p be an analytic polynomial on the unit disk. We obtain a necessary and sufficient condition for Toeplitz operators with the symbol \(\overline z + p\) to be invertible on the Bergman space when all coefficients of p are real numbers. Furthermore, we establish several necessary and sufficient, easy-to-check conditions for Toeplitz operators with the symbol \(\overline z + p\) to be invertible on the Bergman space when some coefficients of p are complex numbers.


Bergman space Toeplitz operator harmonic polynomial symbol invertibility 


47B35 47B65 


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The first author was supported by the Yunnan Natural Science Foundation (Grant No. 201601YA00004). The second author was supported by National Natural Science Foundation of China (Grant No. 11701052), Chongqing Natural Science Foundation (Grant No. cstc2017jcyjAX0373) and the Fundamental Research Funds for the Central Universities (Grant Nos. 106112016CDJRC000080 and 106112017CDJXY100007). The authors thank the reviewers for providing constructive comments and suggestions in improving the contents of this paper. The authors are grateful to Professor Dechao Zheng for various valuable discussion.


  1. 1.
    Ahern P, Čučković Ž. A theorem of Brown-Halmos type for Bergman space Toeplitz operators. J Funct Anal, 2001, 187: 200–210MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Axler S, Čučković Ž. Commuting Toeplitz operators with harmonic symbols. Integral Equations Operator Theory, 1991, 14: 1–12MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benoit B. On the spectrum of Toeplitz operators with quasi-homogeneous symbols. Oper Matrices, 2010, 4: 365–383MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Courant R, John F. Introduction to Calculus and Analysis, Volume 2. New York: Springer, 1989CrossRefzbMATHGoogle Scholar
  5. 5.
    Devinatz A. Toeplitz operators on H 2 spaces. Trans Amer Math Soc, 1964, 112: 304–317MathSciNetzbMATHGoogle Scholar
  6. 6.
    Douglas R. Banach Algebra Techniques in the Theory of Toeplitz Operators. Providence: Amer Math Soc, 1980Google Scholar
  7. 7.
    Douglas R. Banach Algebra Techniques in Operator Theory, 2nd ed. Graduate Texts in Mathematics, vol. 179. New York: Springer, 1998CrossRefzbMATHGoogle Scholar
  8. 8.
    Faour N S. Toeplitz operators on Bergman spaces. Rend Circ Mat Palermo (2), 1986, 35: 221–232MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guo K, Zhao X, Zheng D. The spectra of Toeplitz operators with harmonic polynomial symbols on the Bergman space. ArXiv:1908.05889, 2019Google Scholar
  10. 10.
    Gürdal M, Söhret F. Some results for Toeplitz operators on the Bergman space. Appl Math Comput, 2011, 218: 789–793MathSciNetzbMATHGoogle Scholar
  11. 11.
    Karaev M T. Berezin symbol and invertibility of operators on the functional Hilbert spaces. J Funct Anal, 2006, 238: 181–192MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Luecking D. Inequalities on Bergman spaces. Illinois J Math, 1981, 25: 1–11MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    McDonald G, Sundberg C. Toeplitz operators on the disc. Indiana Univ Math J, 1979, 28: 595–611MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nikolskii N K. Treatise on the Shift Operator: Spectral Function Theory. Berlin-Heidelberg: Springer, 1986CrossRefGoogle Scholar
  15. 15.
    Olsen J F, Reguera M C. On a sharp estimate for Hankel operators and Putnam’s inequality. Rev Mat Iberoam, 2016, 32: 495–510MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rahman Q I, Schmeisser G. Analytic Theory of Polynomials. London Mathematical Society Monographs Series, vol. 26. Oxford: Oxford University Press, 2002zbMATHGoogle Scholar
  17. 17.
    Stroethoff K, Zheng D. Toeplitz and Hankel operators on Bergman spaces. Trans Amer Math Soc, 1992, 329: 773–794MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sundberg C, Zheng D. The spectrum and essential spectrum of Toeplitz operators with harmonic symbols. Indiana Univ Math J, 2010, 59: 385–394MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tolokonnikov V A. Estimates in the Carleson corona theorem, ideals of the algebra H , a problem of S.-Nagy. Zap Nauchn Sem LOMI, 1981, 113: 178–198MathSciNetzbMATHGoogle Scholar
  20. 20.
    Widom H. On the spectrum of a Toeplitz operator. Pacific J Math, 1964, 14: 365–375MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Widom H. Toeplitz operators on H p. Pacific J Math, 1966, 19: 573–582MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wolff T H. Counterexamples to two variants of the Helson-Szegö theorem. J Anal Math, 2002, 88: 41–62MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhao X, Zheng D. Positivity of Toeplitz operators via Berezin transform. J Math Anal Appl, 2014, 416: 881–900MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhao X, Zheng D. The spectrum of Bergman Toeplitz operators with some harmonic symbols. Sci China Math, 2016, 59: 731–740MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhao X, Zheng D. Invertibility of Toeplitz operators via Berezin transforms. J Operator Theory, 2016, 75: 101–121MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhu K. Operator Theory in Function Spaces, 2nd ed. Mathematical Surveys and Monographs, vol. 138. Providence: Amer Math Soc, 2007CrossRefzbMATHGoogle Scholar

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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsChuxiong Normal UniversityChuxiongChina
  2. 2.College of Mathematics and StatisticsChongqing UniversityChongqingChina

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