Skip to main content
SpringerLink
Log in

Subordinacy theory for extended CMV matrices

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We develop subordinacy theory for extended Cantero-Moral-Velázquez (CMV) matrices, i.e., we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the presence or absence of subordinate solutions to the generalized eigenvalue equation. Some corollaries and applications of this result are described as well.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Avila A. Almost reducibility and absolute continuity I. arXiv:1006.0704, 2010

  2. Avila A, Krikorian R. Reducibility or nonuniform hyperbolicity for quasiperiodic Schröding cocycles. Ann of Math (2), 2006, 164: 911–940

    Article  MathSciNet  Google Scholar 

  3. Buschmann D. A proof of the Ishii-Pastur theorem by the method of subordinacy. Univ Iagel Acta Math, 1997, 34: 29–34

    MathSciNet  MATH  Google Scholar 

  4. Damanik D, Erickson J, Fillman J, et al. Quantum intermittency for sparse CMV matrices with an application to quantum walks on the half-line. J Approx Theory, 2016, 208: 59–84

    Article  MathSciNet  Google Scholar 

  5. Damanik D, Killip R, Lenz D. Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity. Comm Math Phys, 2000, 212: 191–204

    Article  MathSciNet  Google Scholar 

  6. Fang L, Damanik D, Guo S. Generic spectral results for CMV matrices with dynamically defined Verblunsky coefficients. J Funct Anal, 2020, 279: 108803

    Article  MathSciNet  Google Scholar 

  7. Geronimo J, Teplyaev A. A difference equation arising from the trigonometric moment problem having random reflection coefficients—an operator-theoretic approach. J Funct Anal, 1994, 123: 12–45

    Article  MathSciNet  Google Scholar 

  8. Gesztesy F, Zinchenko M. Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. J Approx Theory, 2006, 139: 172–213

    Article  MathSciNet  Google Scholar 

  9. Gilbert D. On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints. Proc Roy Soc Edinburgh Sect A, 1989, 112: 213–229

    Article  MathSciNet  Google Scholar 

  10. Gilbert D. On subordinacy and spectral multiplicity for a class of singular differential operators. Proc Roy Soc Edinburgh Sect A, 1998, 128: 549–584

    Article  MathSciNet  Google Scholar 

  11. Gilbert D, Pearson D. On subordinacy and analysis of the spectrum of one-dimensional Schroödinger operators. J Math Anal Appl, 1987, 128: 30–56

    Article  MathSciNet  Google Scholar 

  12. Hadj Amor S. Hölder continuity of the rotation number for quasi-periodic co-cycles in SL(2, ℝ). Comm Math Phys, 2009, 287: 565–588

    Article  MathSciNet  Google Scholar 

  13. Jakšić V, Last Y. A new proof of Poltoratskii’s theorem. J Funct Anal, 2004, 215: 103–110

    Article  MathSciNet  Google Scholar 

  14. Jitomirskaya S, Last Y. Power-law subordinacy and singular spectra I. Half-line operators. Acta Math, 1999, 183: 171–189

    Article  MathSciNet  Google Scholar 

  15. Jitomirskaya S, Last Y. Power-law subordinacy and singular spectra. II. Line operators. Comm Math Phys, 2000, 211: 643–658

    Article  MathSciNet  Google Scholar 

  16. Kac I. On the multiplicity of the spectrum of a second-order differential operator. Soviet Math Dokl, 1962, 3: 1035–1039

    Google Scholar 

  17. Kac I. Spectral multiplicity of a second-order differential operator and expansion in eigenfunction. Izv Akad Nauk SSSR Ser Mat, 1963, 27: 1081–1112. Erratum, Izv Akad Nauk SSSR, 1964, 28: 951–952

    MathSciNet  Google Scholar 

  18. Khan S, Pearson D. Subordinacy and spectral theory for infinite matrices. Helv Phys Acta, 1992, 65: 505–527

    MathSciNet  Google Scholar 

  19. Killip R, Kiselev A, Last Y. Dynamical upper bounds on wavepacket spreading. Amer J Math, 2003, 125: 1165–1198

    Article  MathSciNet  Google Scholar 

  20. Munger P, Ong D. The Hölder continuity of spectral measures of an extended CMV matrix. J Math Phys, 2014, 55: 093507

    Article  MathSciNet  Google Scholar 

  21. Poltoratskiĭ A G. Boundary behavior of pseudocontinuable functions. St Petersburg Math J, 1994, 5: 389–406

    MathSciNet  Google Scholar 

  22. Remling C. Relationships between the m-function and subordinate solutions of second order differential operators. J Math Anal Appl, 1997, 206: 352–363

    Article  MathSciNet  Google Scholar 

  23. Simon B. Analogs of the m-function in the theory of orthogonal polynomials on the unit circle. J Comput Appl Math, 2004, 171: 411–424

    Article  MathSciNet  Google Scholar 

  24. Simon B. On a theorem of Kac and Gilbert. J Funct Anal, 2005, 223: 109–115

    Article  MathSciNet  Google Scholar 

  25. Simon B. Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. Province: Amer Math Soc, 2005

    MATH  Google Scholar 

  26. Simon B. Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. Province: Amer Math Soc, 2005

    MATH  Google Scholar 

  27. Simon B. Aizenman’s theorem for orthogonal polynomials on the unit circle. Constr Approx, 2006, 23: 229–240

    Article  MathSciNet  Google Scholar 

  28. Weidmann J. Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics, vol. 68. New York-Berlin: Springer-Verlag, 1980

    Book  Google Scholar 

Download references

Acknowledgements

The first author was supported by China Scholarship Council (Grant No. 201906330008) and National Natural Science Foundation of China (Grant No. 11571327). The second author was supported by National Science Foundation of USA (Grant No. DMS-1700131) and Alexander von Humboldt Foundation. The third author was supported by the Fundamental Research Grant Scheme from the Malaysian Ministry of Education (Grant No. FRGS/1/2018/STG06/XMU/02/1) and Xiamen University Malaysia Research Fund (Grant No. XMUMRF/2020-C5/IMAT/0011).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China

    Shuzheng Guo

  2. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    Shuzheng Guo & David Damanik

  3. Department of Mathematics, Xiamen University Malaysia, Sepang, Selangor Darul Ehsan, 43900, Malaysia

    Darren C. Ong

Authors
  1. Shuzheng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Damanik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Darren C. Ong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Darren C. Ong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, S., Damanik, D. & Ong, D.C. Subordinacy theory for extended CMV matrices. Sci. China Math. 65, 539–558 (2022). https://doi.org/10.1007/s11425-020-1778-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-020-1778-4

Keywords

MSC(2020)

Search

Navigation