Skip to main content
SpringerLink
Log in

On Two-Dimensional Model Representations of One Class of Commuting Operators

  • Published:
Ukrainian Mathematical Journal Aims and scope

In the work by V. A. Zolotarev, Dokl. Akad. Nauk Arm. SSR, 63, No. 3, 136–140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators {A 1, A 2} ([A 1, A 2] = 0, [A ∗1 , A 2] = 0) such that rank (A 1) I (A 2) I  = 1 (2i(A k ) I  = A k  − A ∗ k , k = 1, 2) and the spectrum of each operator A k , k = 1, 2, is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in L 2 Ω where the domain Ω = [0, a] × [0, b] is a compact set in ℝ2 bounded by the lines x = a and y = b and a decreasing smooth curve L connecting the points (0, b) and (a, 0).

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. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Pitman, Boston, etc., Vol. 1 (1981).

    MATH  Google Scholar 

  2. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd (1965).

  3. M. S. Lifshits, “On the spectral decomposition of linear nonself-adjoint operators,” Mat. Sb., Issue 34 (76), No. 1, 145–198 (1954).

  4. M. S. Livšic and A. A. Yantsevich, Theory of Operator Colligations in Hilbert Spaces, Wiley, New York (1979).

    Google Scholar 

  5. V. A. Zolotarev, Analytic Methods of Spectral Representations of Nonself-Adjoint and Nonunitary Operators [in Russian], MagPress, Kharkov (2003).

    Google Scholar 

  6. V. A. Zolotarev, “On triangular models of the systems of twice commuting operators,” DAN Arm. SSR, 63, No. 3, 136–140 (1976).

    MathSciNet  Google Scholar 

  7. V. A. Zolotarev, “On the structure and triangular models of systems of one class of commuting operators,” Vestnik Khark. Univ., Issue 43, 69–76 (1978).

  8. R. Hatamleh, “Triangular model of systems of linear operators with nilpotent commutators [A,B] and [A ∗ ,B ∗] ,” Vestnik Khark. Univ. Ser. Mat., Prikl. Mat. Mekh., No. 645, 79–84 (2004).

Download references

Author information

Authors and Affiliations

  1. Jadara University, Irbid, Jordan

    R. Hatamleh

  2. Institute of Low-Temperature Physics and Engineering, Ukrainian National Academy of Sciences, Kharkiv, Ukraine

    V. A. Zolotarev

  3. Karazin Kharkiv National University, Kharkiv, Ukraine

    V. A. Zolotarev

Authors
  1. R. Hatamleh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. A. Zolotarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 1, pp. 108–127, January, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hatamleh, R., Zolotarev, V.A. On Two-Dimensional Model Representations of One Class of Commuting Operators. Ukr Math J 66, 122–144 (2014). https://doi.org/10.1007/s11253-014-0916-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0916-9

Keywords

Search

Navigation