Scattering theory for CMV matrices: Uniqueness, Helson-Szegő and strong Szegő theorems. (English) Zbl 1235.47029
The authors 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. The authors 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, the authors characterize the Verblunsky parameters of the CMV matrices as well as the corresponding spectral measures and scattering functions.
Reviewer: Bilender P. Allahverdiev (Isparta)
MSC:
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 39A12 | Discrete version of topics in analysis |
| 47A40 | Scattering theory of linear operators |
| 30H15 | Nevanlinna spaces and Smirnov spaces |
| 30H10 | Hardy spaces |
Keywords:
spectral measure; scattering function; \(\gamma \)-generating pair; Arov regularity; Schur algorithm; Faddeev-Marchenko space; transformation operator; Gelfand-Levitan-Marchenko equationReferences:
| [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) · Zbl 0174.45203 · doi:10.1007/BF01075356 |
| [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) |
| [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) |
| [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) · Zbl 0699.15009 |
| [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) · Zbl 0902.30026 · doi:10.1007/BF01193811 |
| [7] | Arov D., Dym H.: On matricial Nehari problems, J-inner matrix functions and the Muckenhoupt condition. J. Funct. Anal. 181, 227–299 (2001) · Zbl 0980.47014 · doi:10.1006/jfan.2000.3725 |
| [8] | Arov D., Dym H.: Criteria for the strong regularity of J-inner functions and {\(\gamma\)}-generating matrices. J. Math. Anal. Appl. 280, 387–399 (2003) · Zbl 1044.47012 · doi:10.1016/S0022-247X(03)00067-2 |
| [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) · Zbl 1022.42013 · doi:10.1016/S0024-3795(02)00457-3 |
| [11] | Dubovoy, V., Fritzsche, B., Kirstein, B.: Description of Helson-Szego measures in terms of the Schur parameter sequences of associated Schur functions (2010). arXiv:1003.1670v4 [math.FA] · Zbl 1254.30033 |
| [12] | Faddeev L.D.: Properties of the S-matrix of the one-dimensional Schrödinger equation. Trudy Mat. Inst. Steklov. 73, 314–336 (1964) (Russian) · Zbl 0145.46702 |
| [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) · Zbl 0317.62065 |
| [15] | Golinskii B.L., Ibragimov I.A.: On Szego’s limit theorem. Math. USSR Izv. 5, 421–444 (1971) · Zbl 0249.42012 · doi:10.1070/IM1971v005n02ABEH001055 |
| [16] | Golinskii, L., Kheifets, A., Peherstorfer, F., Yuditskii, P.: Faddeev- Marchenko scattering for CMV matrices and the Strong Szego theorem (2008). arXiv:0807.4017v1 [math.SP] |
| [17] | Ibragimov I.A., Rozanov Yu.A.: Gaussian Stochastic Processes. Nauka, Moscow (1970) |
| [18] | Kheifets A.: On regularization of {\(\gamma\)}-generating pairs. J. Funct. Anal. 130, 310–333 (1995) · Zbl 0823.47016 · doi:10.1006/jfan.1995.1072 |
| [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) · Zbl 0990.30025 |
| [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) · Zbl 0497.60033 |
| [21] | Killip R., Nenciu I.: CMV: the unitary analogue of Jacobi matrices. Commun. Pure Appl. Math. 60, 1148–1188 (2007) · Zbl 1128.15012 · doi:10.1002/cpa.20160 |
| [22] | Marchenko V.: Sturm-Liouville Operators and Applications. Birkhäuser Verlag, Basel (1986) · Zbl 0592.34011 |
| [23] | Marchenko V.: Nonlinear Equations and Operator Algebras. Mathematics and its Applications (Soviet Series), vol. 17. D. Reidel Publishing Co., Dordrecht (1988) · Zbl 0644.47053 |
| [24] | Nikolskii N.: Treatise on the Shift Operator. Springer, Berlin (1986) |
| [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) · Zbl 0458.47022 |
| [26] | Peller V.: Hankel Operators and Their Applications. Springer, Berlin (2003) · Zbl 1030.47002 |
| [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)] · Zbl 0678.46019 |
| [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) · Zbl 1082.42020 |
| [30] | Simon B.: CMV matrices: five years after. J. Comput. Appl. Math. 208, 120–154 (2007) · Zbl 1125.15027 · doi:10.1016/j.cam.2006.10.033 |
| [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) · Zbl 1198.47033 |
| [32] | Widom H.: Asymptotic behavior of block Toeplitz matrices and determinants, II. Adv. Math. 21, 1–29 (1976) · Zbl 0344.47016 · doi:10.1016/0001-8708(76)90113-4 |
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