Journal of Mathematical Sciences

, Volume 172, Issue 2, pp 185–194 | Cite as

On Toeplitz operators with unimodular symbols: left invertibility and similarity to isometries

  • M. F. Gamal’

We consider Toeplitz operators with unimodular symbols on the Hardy space H 2 on the unit circle. Examples of inner functions θ andareal function w such that ||w|| = π/2 and the Toeplitz operator with symbol θe iw is not left-invertible are given. We also study Toeplitz operators that are similar to isometries. Bibliography: 28 titles.


Russia Unit Circle Real Function Hardy Space Toeplitz Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Berovii, Operator Theory and Arithmetic in H , Amer. Math. Soc. Math. Surveys and Monographs, 26 (1988).Google Scholar
  2. 2.
    H. Berovii, “Notes on invariant subspaces,” Bull. Amer. Math. Soc., 23, 1–36 (1990).CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Böttcher and S. M. Grudsky, “Toeplitz operators with discontinuous symbols: phenomena beyond piece-wise continuity,” Oper. Theory Adv. Appl., 90, 55–118 (1996).Google Scholar
  4. 4.
    A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer (1990).Google Scholar
  5. 5.
    D. N. Clark, “On a similarity theory for rational Toeplitz operators,” J. Reine Angew. Math., 320, 6–31 (1980).MATHMathSciNetGoogle Scholar
  6. 6.
    D. N. Clark, “On Toeplitz operators with loops,” J. Operator Theory, 4, 37–54 (1980).MATHMathSciNetGoogle Scholar
  7. 7.
    D. N. Clark, “On Toeplitz operators with unimodular symbols,” Oper. Theory Adv. Appl., 24, 59–68 (1987).Google Scholar
  8. 8.
    D. N. Clark, “Perturbation and similarity of Toeplitz operators,” Oper. Theory Adv. Appl., 48, 235–243 (1990).Google Scholar
  9. 9.
    D. N. Clark and J. H. Morrel, “On Toeplitz operators and similarity,” Amer. J. Math., 100, 973–986 (1978).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. L. Duren, Theory of HP Spaces, Academic Press (1970).Google Scholar
  11. 11.
    V. B. Dybin and S. M. Grudsky, Introduction to the Theory of Toeplitz Operators With Infnite Index, Oper. Theory Adv. Appl., 137 (2002).Google Scholar
  12. 12.
    M. F. Gamal’, “On Toeplitz operators similar to unilateral shift,” Zap. Nauhn. Sem. POMI, 345, 85–104 (2007).Google Scholar
  13. 13.
    M. F. Gamal’, “On Toeplitz operators similar to isometries,” J. Operator Theory, 59, 3–28(2008).MathSciNetGoogle Scholar
  14. 14.
    M. F. Gamal’, “On contrations that are quasiaffine transforms of unilateral shifts,” Acta Sci. Math. (Szeged), 74, 757–767 (2008).MathSciNetGoogle Scholar
  15. 15.
    R. Goor, “On Toeplitz operators which are contractions,” Proc. Amer. Math. Soc., 34, 191–192 (1972).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    V. V. Peller, “When is a function of a Toeplitz operator close to a Toeplitz operator?” Oper. Theory Adv. Appl., 42, 59–85 (1989).MathSciNetGoogle Scholar
  17. 17.
    V. V. Peller, Hankel Operators and Their Applications, Springer Monographs in Math. (2003).Google Scholar
  18. 18.
    V. V. Peller and S. V. Khrushev, “Hankel operator, best approximation, and stationary Gaussian processes,” Usp. Mat. Nauk, 37, 53–124 (1982).MATHGoogle Scholar
  19. 19.
    H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer (1973).Google Scholar
  20. 20.
    M. Rosenblum and J. Rovnyak, Hardy Glasses and Operator Theory, Oxford Math. Monogr. (1985).Google Scholar
  21. 21.
    J. Rovnyak, “On the theory of unbounded Toeplitz operators,” Pacific J. Math., 31, 481–496 (1969).MATHMathSciNetGoogle Scholar
  22. 22.
    D. Sarason, “Approximation of piecewise continuous functions by quotients of bounded analytic funtions,” Ganad. J. Math., 24, 642–657 (1972).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Spaces, North Holland, Amsterdam (1970).MATHGoogle Scholar
  24. 24.
    K. Takahashi, “Injection of unilateral shifts into contractions,” Acta Sci. Math. (Szeged), 57, 263–276 (1993).MATHMathSciNetGoogle Scholar
  25. 25.
    D. Wang, “Similarity theory of smooth Toeplitz operators,” J. Operator Theory, 12, 319–330 (1984).MATHMathSciNetGoogle Scholar
  26. 26.
    D. V. Yakubovih, “Riemann surface models of a Toeplitz operator,” Oper. Theory Adv. Appl., 42, 305–415 (1989).Google Scholar
  27. 27.
    D. V. Yakubovih, “On the spectral theory of Toeplitz operators with a smooth symbol,” Algebra Analiz, 3, 208–226(1991).Google Scholar
  28. 28.
    D. V. Yakubovich, “Dual piecewise analytic bundle shift models of linear operators,” J. Funct. Anal., 136, 294–330(1996).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

Personalised recommendations