Skip to main content
SpringerLink
Log in

Almost Invariant Subspaces of the Shift Operator on Vector-Valued Hardy Spaces

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

Abstract

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar–Gallardo–Partington. Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.

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.

Similar content being viewed by others

References

  1. Androulakis, G., Popov, A.I., Tcaciuc, A., Troitsky, V.G.: Almost invariant half-spaces of operators on Banach spaces. Integral Equ. Oper. Theory 65(4), 473–484 (2009)

    Article  MathSciNet  Google Scholar 

  2. Benhida, C., Timotin, D.: Finite rank perturbations of contractions. Integral Equ. Oper. Theory 36(3), 253–268 (2000)

    Article  MathSciNet  Google Scholar 

  3. Beurling, A.: On two problems concerning linear transformations in Hilbert space. Acta Math. 81, 239–255 (1949)

    Article  MathSciNet  Google Scholar 

  4. Chalendar, I., Chevrot, N., Partington, J.R.: Nearly invariant subspaces for backwards shifts on vector-valued Hardy spaces. J. Oper. Theory 63(2), 403–415 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Chalendar, I., Gallardo-Gutiérrez, E. A., Partington, J. R.: A Beurling Theorem for almost-invariant subspaces of the shift operator. arXiv:1905.06652

  6. Chevrot, N.: Kernel of vector-valued Toeplitz operators. Integral Equ. Oper. Theory 67(1), 57–78 (2010)

    Article  MathSciNet  Google Scholar 

  7. Fricain, E., Mashreghi, J.: The Theory of H(b) Spaces. Vol. 1. New Mathematical Monographs, vol. 20. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  8. Fricain, E., Mashreghi, J.: The theory of H(b) spacesVol2, New Mathematical Monographs, vol. 21. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  9. Hitt, D.: Invariant subspaces of \(H^2\) of an annulus. Pac. J. Math. 134(1), 101–120 (1988)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Nikolski, N.: Operators, Functions, and Systems: An Easy Reading. Vol 1. Hardy, Hankel, and Toeplitz. Mathematical Surveys and Monographs, vol. 92. American Mathematical Society, Providence (2002)

    Google Scholar 

  12. Partington, J.R.: Linear Operators and Linear Systems. An Analytical Approach to Control Theory. London Mathematical Society Student Texts, vol. 60. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  13. Popov, A., Tcaciuc, A.: Every operator has almost-invariant subspaces. J. Funct. Anal. 265(2), 257–265 (2013)

    Article  MathSciNet  Google Scholar 

  14. Sarason, D.: Nearly Invariant Subspaces of the Backward Shift. Contributions to Operator Theory and ItsApplications (Mesa, AZ, 1987). Operator Theory: Advances and Applications, vol. 35, pp. 481–493. Birkhäuser, Basel (1988)

    Google Scholar 

  15. Sirotkin, G., Wallis, B.: The structure of almost-invariant half-spaces for some operators. J. Funct. Anal. 267(7), 2298–2312 (2014)

    Article  MathSciNet  Google Scholar 

  16. Sz.-Nagy, B., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. North Holland, Amsterdam (1970)

    MATH  Google Scholar 

  17. Tcaciuc, A.: The invariant subspace problem for rank one perturbations. Duke Math. J. 168(8), 1539–1550 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are extremely grateful to Dr. Bata Krishna Das for many fruitful discussions and his valuable comments. We would also like to thank Prof. Jaydeb Sarkar for introducing this area to us. The research of the first named author is supported by the Mathematical Research Impact Centric Support (MATRICS) grant, File No: MTR/2019/000640, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. The second and the third named author gratefully acknowledge the support provided by IIT Guwahati, Government of India.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Arup Chattopadhyay, Soma Das & Chandan Pradhan

Authors
  1. Arup Chattopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Soma Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chandan Pradhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arup Chattopadhyay.

Additional information

Publisher's Note

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

This paper is published in the form submitted to IEOT on January 14, 2020. A related paper by R. O’Loughlin was submitted to arxiv some months later, see arXiv:2005.00378 [math.FA].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chattopadhyay, A., Das, S. & Pradhan, C. Almost Invariant Subspaces of the Shift Operator on Vector-Valued Hardy Spaces. Integr. Equ. Oper. Theory 92, 52 (2020). https://doi.org/10.1007/s00020-020-02612-6

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-020-02612-6

Keywords

Mathematics Subject Classification

Search

Navigation