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Scattering Theory for CMV Matrices: Uniqueness, Helson–Szegő and Strong Szegő Theorems

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Abstract

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.

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References

  1. Adamjan V., Arov D., Kreĭn M.: Infinite Hankel matrices and generalized problems of Caratheodory-Fejér and F. Riesz. Funkcional Anal. i Priložen 2(1), 1–19 (1968) (Russian)

    Article  Google Scholar 

  2. Adamjan V., Arov D., Kreĭ n M.: Infinite Hankel matrices and generalized Carathéodory-Fejér and I. Schur problems. Funkcional Anal. i Priložen. 2(4), 1–17 (1968) (Russian)

    Article  Google Scholar 

  3. Adamjan V., Arov D., Kreĭn M.: Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem. Mat. Sb. (N.S.) 86(128), 34–75 (1971) (Russian)

    MathSciNet  Google Scholar 

  4. Arov, D.Z.: Regular and singular J-inner matrix functions and corresponding extrapolation problems. Funktsional Anal. i Prilozhen. 22(1), 57–59 (1988) (Russian). Translation in Funct. Anal. Appl. 22(1), 46–48 (1988)

  5. Arov D.Z.: Regular J-inner matrix-functions and related continuation problems. In: Helson, H., Sz.-Nagy, B., Vasilescu, F.-H. (eds) Linear Operators in Functions Spaces, vol. OT 43, pp. 63–87. Birkhäuser-Verlag, Basel (1990)

    Google Scholar 

  6. Arov D.Z., Dym H.: J-inner matrix functions, interpolation and inverse problems for canonical systems. I. Foundations. Integral Equ. Oper. Theory 29(4), 373–454 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arov D., Dym H.: On matricial Nehari problems, J-inner matrix functions and the Muckenhoupt condition. J. Funct. Anal. 181, 227–299 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Arov D., Dym H.: Criteria for the strong regularity of J-inner functions and γ-generating matrices. J. Math. Anal. Appl. 280, 387–399 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ball, J., Kheifets, A.: The inverse commutant lifting problem: characterization of associated Redheffer linear-fractional maps (2010). arXiv:1004.0447v1 [math.FA]

  10. Cantero M.J., Moral L., Velázquez L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dubovoy, V., Fritzsche, B., Kirstein, B.: Description of Helson-Szegő measures in terms of the Schur parameter sequences of associated Schur functions (2010). arXiv:1003.1670v4 [math.FA]

  12. Faddeev L.D.: Properties of the S-matrix of the one-dimensional Schrödinger equation. Trudy Mat. Inst. Steklov. 73, 314–336 (1964) (Russian)

    MATH  MathSciNet  Google Scholar 

  13. Garnett, J.B.: Bounded Analytic Functions, revised 1st edn. Graduate Texts in Mathematics, vol. 236, xiv+459 pp. Springer, New York (2007)

  14. Golinskii B.L.: On asymptotic behavior of the prediction error. Probab. Theory Appl. 19(4), 224–239 (1974) (Russian)

    Google Scholar 

  15. Golinskii B.L., Ibragimov I.A.: On Szegő’s limit theorem. Math. USSR Izv. 5, 421–444 (1971)

    Article  Google Scholar 

  16. Golinskii, L., Kheifets, A., Peherstorfer, F., Yuditskii, P.: Faddeev- Marchenko scattering for CMV matrices and the Strong Szegő theorem (2008). arXiv:0807.4017v1 [math.SP]

  17. Ibragimov I.A., Rozanov Yu.A.: Gaussian Stochastic Processes. Nauka, Moscow (1970)

    Google Scholar 

  18. Kheifets A.: On regularization of γ-generating pairs. J. Funct. Anal. 130, 310–333 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kheifets, A.: Nehari’s interpolation problem and exposed points of the unit ball in the Hardy space H 1. In: Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), pp. 145–151, Israel Mathematical Conference Proceedings, vol. 11. Bar-Ilan University, Ramat Gan (1997)

  20. Khrushchev, S., Peller, V.: Hankel operators, best approximations, and stationary Gaussian processes. Uspekhi Mat. Nauk 37(1), 53–124 (1982). English Transl. in Russ. Math. Surv. 37(1), 61–144 (1982)

  21. Killip R., Nenciu I.: CMV: the unitary analogue of Jacobi matrices. Commun. Pure Appl. Math. 60, 1148–1188 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Marchenko V.: Sturm-Liouville Operators and Applications. Birkhäuser Verlag, Basel (1986)

    MATH  Google Scholar 

  23. Marchenko V.: Nonlinear Equations and Operator Algebras. Mathematics and its Applications (Soviet Series), vol. 17. D. Reidel Publishing Co., Dordrecht (1988)

    Google Scholar 

  24. Nikolskii N.: Treatise on the Shift Operator. Springer, Berlin (1986)

    Google Scholar 

  25. Peller, V.: Hankel operators of class S p and their applications (rational approximation, Gaussian processes, the problem of majorizing operators). Mat. Sbornik 113, 538–581 (1980). English Transl. in Math. USSR Sbornik 41, 443–479 (1982)

    Google Scholar 

  26. Peller V.: Hankel Operators and Their Applications. Springer, Berlin (2003)

    MATH  Google Scholar 

  27. Sarason, D.: Exposed points in H 1. Operaor Theory: Advances and Applications, vol. 41, pp. 485–496. Birkhäuser, Basel (1989) [vol. 48, pp. 333–347 (1990)]

  28. Schur, I.: Über Potenzreihn, die im Innern des Einheitskreises beschränkt sind, I, II. J. Reine Angew. Math. 147, 205–232 (1917) [148, 122–145 (1918)]

  29. Simon B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory; Part 2: Spectral Theory. AMS Colloquium Series. AMS, Providence (2005)

    Google Scholar 

  30. Simon B.: CMV matrices: five years after. J. Comput. Appl. Math. 208, 120–154 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Volberg A., Yuditskii P.: Remarks on Nehari’s problem, matrix A 2 condition, and weighted bounded mean oscillation. Am. Math. Soc. Transl. 226(2), 239–254 (2009)

    MathSciNet  Google Scholar 

  32. Widom H.: Asymptotic behavior of block Toeplitz matrices and determinants, II. Adv. Math. 21, 1–29 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Division, Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., Kharkov, 61103, Ukraine

    L. Golinskii

  2. Department of Mathematics, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    A. Kheifets

  3. Institute for Analysis, Johannes Kepler University of Linz, 4040, Linz, Austria

    F. Peherstorfer & P. Yuditskii

Authors
  1. L. Golinskii
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  2. A. Kheifets
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  3. F. Peherstorfer
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  4. P. Yuditskii
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Corresponding author

Correspondence to A. Kheifets.

Additional information

F. Peherstorfer: Deceased.

The work of A. Kheifets was partially supported by the University of Massachusetts Lowell Research and Scholarship Grant, project number: H50090000000010. The work of F. Peherstorfer and P. Yuditskii was partially supported by the Austrian Science Found FWF, project number: P20413-N18

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Golinskii, L., Kheifets, A., Peherstorfer, F. et al. Scattering Theory for CMV Matrices: Uniqueness, Helson–Szegő and Strong Szegő Theorems. Integr. Equ. Oper. Theory 69, 479–508 (2011). https://doi.org/10.1007/s00020-010-1859-7

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  • DOI: https://doi.org/10.1007/s00020-010-1859-7

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