On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

• Autorzy: Ianovich E.

Identyfikatory

DOI

10.7494/OpMath.2018.38.5.699

• Języki publikacji: EN

Abstrakty

EN

In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator A by a sequence of operators An with absolutely continuous spectrum on a given interval [a, b] which converges to A in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator A spectrum on the finite interval [a, b] and the condition for that the corresponding spectral density belongs to the class Lp[a,b] (p ≥ 1). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant C > 0 and a positive function g(x) ∈ Lp[a, b] (p ≥ 1).such that for all n sufficiently large and almost all [formula] the estimate [formula] holds, where Pn(x) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and bn is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on [a, b] and for the corresponding spectral density ƒ (x) we have ƒ (x) ∈ Lp[a,b].

Słowa kluczowe

EN

self-adjoint operators absolutely continuous spectrum; equi-absolute continuity; spectral density; Jacobi matrices

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2018
• Tom: Vol. 38, no. 5
• Strony: 699--718
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Ianovich E.

• Saint Petersburg, Russia

Bibliografia

• [1] N.I. Akhiejzer, The Classical Moment Problem, New-York, Hafner, 1965.
• [2] N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications, Inc. New York, 1993.
• [3] A. Aptekarev, J. Geronimo, Measures for orthogonal polynomials with unbounded recurrence coefficients, arXiv:1408.5349
• [4] A. Aptekarev, J. Geronimo, Measures for orthogonal polynomials with unbounded recurrence coefficients, J. Approx. Theory 207 (2016), 339-347.
• [5] S. Clark, A spectral analysis for self-adjoint operators generated by a class of second order difference equations, J. Math. Anal. Appl. 197 (1996) 1, 267-285.
• [6] J. Dombrowski, Cyclic operators, commutators, and absolutely continuous measures, Proc. Amer. Math. Soc. 100 (1987) 3, 457-463.
• [7] J. Geronimo, W. Van Assche, Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991) 3, 341-371.
• [8] E.A. Ianovich, Jacobi matrices: continued fractions, approximation, spectrum, arXiv:1707.04695
• [9] J. Janas, M. Moszynski, Alternative approaches to the absolute continuity of Jacobi matrices with monotonie weights, Integral Equations Operator Theory 43 (2002), 397-416.
• [10] J. Janas, S. Naboko, Jacobi matrices with power-like weights-grouping in blocks approach, J. Funct. Anal. 166 (1999), 218-243.
• [11] J. Janas, S. Naboko, Asymptotics of generalized eigenvectors for unbounded Jacobi matrices with power-like weights, Pauli matrices commutation relations and Cesaro averaging, Oper. Theory Adv. Appl. 117 (2000), 165-186.
• [12] J. Janas, S. Naboko, Multithreshold spectral phase transitions for a class of Jacobi matrices, Oper. Theory Adv. Appl. 124 (2001), 267-285.
• [13] A. Mate, P. Nevai, Orthogonal polynomials and absolutely continuous measures, [in:] O.K. Chui et al., Approximation IV, Academic Press, New York, 1983, pp. 611-617.
• [14] M. Moszynski, Spectral properties of some Jacobi matrices with double weights, J. Math. Anal. Appl. 280 (2003), 400-412.
• [15] I.P. Natanson, Theory of Functions of a Real Variable, vol. 1, Frederick Ungar Publishing Company, New York, 1964.
• [16] I.I. Privalov, Boundary Properties of Analytic Functions, Gostehizdat, Moscow, 1950. [17] V.I. Smirnov, Course of Higher Mathematics, vol. 5, Gostehizdat, Moscow, 1959.
• [18] G. Stolz, Spectral theory for slowly oscillating potentials, I. Jacobi matrices, Manuscripta Math. 84 (1994), 245-260.
• [19] G. Swiderski, Periodic perturbations of unbounded Jacobi matrices II: Formulas for density, J. Approx. Theory 216 (2017), 67-85.
• [20] G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, Rhode Island, 1939.
• [21] R. Szwarc, Absolute continuity of certain unbounded Jacobi matrices. Advanced Problems in Constructive Approximation, [in:] M.D. Buhmann, D.H. Mache (eds.), International Series of Numerical Mathematics, vol. 142, 2003, pp. 255-262.
• [22] P. Turan, On the zeros of the polynomials of Legendre, Casopis Pest Mat. Fys. 75 (1950), 113-122.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).