Skip to main content
SpringerLink
Log in

Volterra-Type Discrete Integral Equations and Spectra of Non-self-adjoint Jacobi Operators

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

Abstract

We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb–Thirring inequality due to Hansmann and Katriel. The spectral enclosure for such operators is 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. Bögli, S., Štampach, F.: On Lieb-Thirring inequalities for one-dimensional non-self-adjoin Jacobi and Schrödinger operators. J. Spectr. Theory 11, 1391–1413 (2021)

    Article  MathSciNet  Google Scholar 

  2. Borichev, A., Golinskii, L., Kupin, S.: A Blaschke-type condition and its application to complex Jacobi matrices. Bull. Lond. Math. Soc. 41, 117–123 (2009)

    Article  MathSciNet  Google Scholar 

  3. Borichev, A., Golinskii, L., Kupin, S.: On zeros of analytic functions satisfying non-radial growth conditions. Rev. Mat. Iberoam. 34, 1153–1176 (2018)

    Article  MathSciNet  Google Scholar 

  4. Demuth, M., Hansmann, M., Katriel, G.: Eigenvalues of non-selfadjoint operators: a comparison of two approaches. In: Mathematical physics, spectral theory and stochastic analysis, Oper. Theory Adv. Appl., pp. 107–163, Birkhäuser/Springer (2013)

  5. Egorova, I., Golinskii, L.: On the location of the discrete spectrum for complex Jacobi matrices. Proc. Am. Math. Soc. 133, 3635–3641 (2005)

    Article  MathSciNet  Google Scholar 

  6. Favorov, S., Golinskii, L.: A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators. In: Advances in Mathematical Sciences, Linear and Complex Analysis, pp. 37–47 (2009)

  7. Frank, R.: The Lieb–Thirring inequalities: recent results and open problems. In: Nine Mathematical Challenges, Proc. of Symposia in Pure Mathematics, 104, pp. 45–86 (2021)

  8. Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. AMS, Providence RI (1969)

    MATH  Google Scholar 

  9. Hansmann, M., Katriel, G.: Inequalities for the eigenvalues of non-selfadjoint Jacobi operators. Complex Anal. Oper. Theory 5, 197–218 (2011)

    Article  MathSciNet  Google Scholar 

  10. Ibrogimov, O., Štampach, F.: Spectral enclosures for non-self-adjoint discrete Schrödinger operators. Integral Equ. Oper. Theory 91, 53 (2019). https://doi.org/10.1007/s00020-019-2553-z

    Article  MATH  Google Scholar 

  11. Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158, 253–321 (2003)

    Article  MathSciNet  Google Scholar 

  12. Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. AMS, Providence RI (2000)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. B. Verkin Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, 47 Nauky Ave., Kharkiv, 61103, Ukraine

    L. Golinskii

Authors
  1. L. Golinskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Golinskii.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Golinskii, L. Volterra-Type Discrete Integral Equations and Spectra of Non-self-adjoint Jacobi Operators. Integr. Equ. Oper. Theory 93, 63 (2021). https://doi.org/10.1007/s00020-021-02679-9

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-021-02679-9

Keywords

Mathematics Subject Classification

Search

Navigation