Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 287–300 | Cite as

Some results for operators on a model space

  • Mehmet Gürdal
  • Ulaş Yamanci
  • Mübariz Garayev
Research Article


We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K θ = H2ΘθH2 is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H2-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.


Reproducing kernel Hilbert space (RKHS) reproducing kernel Berezin symbol Berezin number truncated Toeplitz operator model operator Toeplitz operator 


47A35 47B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank the referees for their useful and constructive remarks and suggestions which improved the presentation of the paper. This work was supported by TüBA through Young Scientist Award Program (TüBA-GEB_IP/2015). Also, the third author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saudi University for funding this Research group No. RGP-VPP-323.


  1. 1.
    Bayart F, Matheron E. Dynamics of Linear Operators. Cambridge Univ Press, 2009CrossRefzbMATHGoogle Scholar
  2. 2.
    Berezin F A. Covariant and contravariant symbols for operators. Math USSR-Izv, 1972, 6: 1117–1151CrossRefzbMATHGoogle Scholar
  3. 3.
    Berezin F A. Quantization. Math USSR-Izv, 1974, 8: 1109–1163CrossRefzbMATHGoogle Scholar
  4. 4.
    Chalendar I, Fricain E, Gürdal M, Karaev M T. Compactness and Berezin symbols. Acta Sci Math (Szeged), 2012, 78: 315–329MathSciNetzbMATHGoogle Scholar
  5. 5.
    Clark D. One dimensional perturbations of restricted shifts. J Anal Math, 1972, 25: 169–191MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Englis M. Toeplitz operators and the Berezin transform on H2: Linear Algebra Appl, 1995, 223-224: 171–204MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gürdal M, Şohret F. Some results for Toeplitz operators on the Bergman space. Appl Math Comput, 2011, 218: 789–793MathSciNetzbMATHGoogle Scholar
  8. 8.
    Karaev M T. Berezin symbols and Schatten-von Neumann classes. Math Notes, 2002, 72: 185–192MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Karaev M T. Berezin symbol and invertibility of operators on the functional Hilbert spaces. J Funct Anal, 2006, 238: 181–192MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karaev M T. On the Riccati equations. Monatsh Math, 2008, 155: 161–166MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nikolski N K. Treatise on the Shift Operator. Heidelberg: Springer-Verlag, 1986CrossRefGoogle Scholar
  12. 12.
    Sarason D. Algebraic properties of truncated Toeplitz operators. Oper Matrices, 2007, 1: 491–526MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang X F, Xia J, Cao G F. Some properties of Toeplitz operators on Dirichlet space with respect to Berezin type transform. Sci Sin Math, 2012, 42(11): 1159–1170CrossRefGoogle Scholar
  14. 14.
    Zhu K. Operator Theory in Function Spaces. 2nd ed. Providence: Amer Math Soc, 2007CrossRefzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Mehmet Gürdal
    • 1
  • Ulaş Yamanci
    • 2
  • Mübariz Garayev
    • 3
  1. 1.Department of MathematicsSuleyman Demirel UniversityIspartaTurkey
  2. 2.Department of StatisticsSuleyman Demirel UniversityIspartaTurkey
  3. 3.Department of Mathematics, College of ScienceKing Saud UniversityRiyadhSaudi Arabia

Personalised recommendations