Skip to main content
SpringerLink
Log in

Near invariance and kernels of Toeplitz operators

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

This paper makes a systematic study of kernels of Toeplitz operators on scalar and vector-valued H p spaces (for 1 < p < ∞). The property of near invariance of a kernel for the backward shift is analysed and shown to hold in increased generality. In the scalar case, and in some vectorial cases, the existence of a minimal kernel containing a given function is established, and a symbol for a corresponding Toeplitz operator is determined; thus, for rational symbols, its dimension can be easily calculated. It is shown that every Toeplitz kernel in H p is the minimal kernel for some function lying in it.

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. C. Benhida, M. C. Câmara, and C. Diogo, Some properties of the kernel and the cokernel of Toeplitz operators with matrix symbols, Linear Algebra Appl. 432 (2010), 307–317.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  3. N. Chevrot, Kernel of vector-valued Toeplitz operators, Integral Equations Operator Theory 67 (2010), 57–78.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. L. Duren, Theory of H p Spaces. Dover, New York, 2000.

    Google Scholar 

  5. K. M. Dyakonov, Kernels of Toeplitz operators via Bourgain’s factorization theorem, J. Funct. Anal. 170 (2000), 93–106.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Hayashi, The kernel of a Toeplitz operator, Integral Equations Operator Theory 9 (1986), 588–591.

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Hayashi, Classification of nearly invariant subspaces of the backward shift, Proc. Amer. Math. Soc. 110 (1990), 441–448.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Hitt, Invariant subspaces of H 2 of an annulus, Pacific J. Math. 134 (1988), 101–120.

    Article  MATH  MathSciNet  Google Scholar 

  9. N. K. Nikolski, Operators, Functions, and Systems: an Easy Reading. Vol. 1. Hardy, Hankel, and Toeplitz, American Mathematical Society, Providence, RI, 2002.

    Google Scholar 

  10. D. Sarason, Nearly invariant subspaces of the backward shift, Contributions to Operator Theory and its Applications, Birkhäuser Verlag, Basel, 1988, pp. 481–493.

    Google Scholar 

  11. D. Sarason, Kernels of Toeplitz operators, Toeplitz Operators and Related Topics, Birkhäuser Verlag, Basel, 1994, pp. 153–164.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, 1049-001, Lisboa, Portugal

    M. Cristina Câmara

  2. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Jonathan R. Partington

Authors
  1. M. Cristina Câmara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonathan R. Partington
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Cristina Câmara.

Additional information

This research was partially supported by FCT (Portugal) through program POCTI/FEDER and Project PEst-OE/EEI/LA0009/2013.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Câmara, M.C., Partington, J.R. Near invariance and kernels of Toeplitz operators. JAMA 124, 235–260 (2014). https://doi.org/10.1007/s11854-014-0031-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-014-0031-8

Keywords

Search

Navigation