Skip to main content
SpringerLink
Log in

Some Convergence Theorems for Operator Sequences

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

Let A,  T, and B be bounded linear operators on a Banach space. This paper is concerned mainly with finding some necessary and sufficient conditions for convergence in operator norm of the sequences \(\left\{ A^{n}TB^{n}\right\} \) and \(\left\{ \frac{1}{n}\sum _{i=0}^{n-1}A^{i}TB^{i} \right\} \). These results are applied to the Toeplitz, composition, and model operators. Some related problems are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Axler, S., Chang, S.-Y.A., Sarason, D.: Products of Toeplitz operators. Integr. Equ. Oper. Theory 1, 285–309 (1978)

    Article  MathSciNet  Google Scholar 

  2. Barria, J., Halmos, P.R.: Asymptotic Toeplitz operators. Trans. Am. Math. Soc. 273, 621–630 (1982)

    Article  MathSciNet  Google Scholar 

  3. Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North Holland, Amsterdam (1988)

    MATH  Google Scholar 

  4. Brown, A., Halmos, P.R.: Algebraic properties of Toeplitz operators. J. R. Angew. Math. 213, 89–102 (1963/1964)

  5. Conway, J.B.: A Course in Functional Analysis. Graduate Texts in Mathematics. Springer, Berlin (1985)

    Book  Google Scholar 

  6. Douglas, R.G.: Banach Algebra Techniques in Operator Theory. Academic Press, New York (1972)

    MATH  Google Scholar 

  7. Dunford, N., Schwartz, J.T.: Linear Operators III. Mir, Moscow (1974). (Russian)

    MATH  Google Scholar 

  8. Eisner, T.: Stability of Operators and Operator Semigroups. Operator Theory: Advances and Applications, vol. 209. Birkhäuser, Basel (2010)

    Book  Google Scholar 

  9. Feintuch, A.: On asymptotic Toeplitz and Hankel operators. In the Gohberg anniversary collection. Oper. Theory Adv. Appl. 41, 241–254 (1989)

    Google Scholar 

  10. Gamelin, T., Garnett, J.: Uniform approximation to bounded analytic functions. Rev. Un. Mat. Argentina 25, 87–94 (1970)

    MathSciNet  MATH  Google Scholar 

  11. Jung, I.B., Ko, E., Pearcy, C.: A note on the spectral mapping theorem. Kyungpook Math. J. 47, 77–79 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Katznelson, Y., Tzafriri, L.: On power bounded operators. J. Funct. Anal. 68, 313–328 (1986)

    Article  MathSciNet  Google Scholar 

  13. Kellay, K., Zarrabi, M.: Compact operators that commute with a contraction. Integr. Equ. Oper. Theory 65, 543–550 (2009)

    Article  MathSciNet  Google Scholar 

  14. Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)

    Book  Google Scholar 

  15. Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. Clarendon Press, Oxford (2000)

    MATH  Google Scholar 

  16. Lumer, G., Rosenblum, M.: Linear operator equations. Proc. Am. Math. Soc. 10, 32–41 (1959)

    Article  MathSciNet  Google Scholar 

  17. Martinez-Avendaño, R.A.: Essentially Hankel operators. J. Lond. Math. Soc. 66, 741–752 (2002)

    Article  MathSciNet  Google Scholar 

  18. Martinez-Avendaño, R.A., Rosenthal, P.: An Introduction to Operators on the Hardy-Hilbert Space. Graduate Texts in Mathematics, vol. 237. Springer, Berlin (2007)

    MATH  Google Scholar 

  19. Mustafayev, H.S.: Asymptotic behavior of polynomially bounded operators. C. R. Math. Acad. Sci. Paris 348, 517–520 (2010)

    Article  MathSciNet  Google Scholar 

  20. Mustafayev, H.S.: Growth conditions for conjugate orbits of operators on Banach spaces. J. Oper. Theory 74, 281–306 (2015)

    Article  MathSciNet  Google Scholar 

  21. Nagy, B.S., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. Mir, Moscow (1970). (Russian)

    MATH  Google Scholar 

  22. Nagy, B.S., Foiaş, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space. Springer, New York (2010)

    Book  Google Scholar 

  23. Nazarov, F., Shapiro, J.H.: On the toeplitzness of composition operators. Complex Var. Elliptic Equ. 52, 193–210 (2007)

    Article  MathSciNet  Google Scholar 

  24. Nikolski, N.K.: Treatise on the Shift Operator. Nauka, Moscow (1980). (Russian)

    Google Scholar 

  25. Nikolski, N.K.: Operators, Functions, and Systems: An Easy Reading. Vol. I: Hardy, Hankel, and Toeplitz, AMS, Mathematical Surveys and Monographs, vol. 92. American Mathematical Society, Providence (2002)

    Google Scholar 

  26. Volberg, A.L.: Two remarks concerning the theorem of S. Axler, S.-Y. A. Chang, and D. Sarason. J. Oper. Theory 7, 209–218 (1982)

    MathSciNet  MATH  Google Scholar 

  27. Zarrabi, M.: Some results of Katznelson–Tzafriri type. J. Math. Anal. Appl. 397, 109–118 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the referee for his helpful remarks, suggestions, and substantial improvements to the paper. The author was supported by TÜBİTAK (The Scientific and Technological Research Council of Turkey) 1001 Project MFAG No: 118F410.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Van Yuzuncu Yil University, Van, Turkey

    Heybetkulu Mustafayev

Authors
  1. Heybetkulu Mustafayev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heybetkulu Mustafayev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was completed with the support of our TeX-pert.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mustafayev, H. Some Convergence Theorems for Operator Sequences. Integr. Equ. Oper. Theory 92, 36 (2020). https://doi.org/10.1007/s00020-020-02591-8

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-020-02591-8

Keywords

Mathematics Subject Classification

Search

Navigation