Abstract
We consider a bounded Jacobi operator acting in the space l 2(ℕ). We supplement the spectral measure of this operator by a set of finitely many discrete masses (on the real axis outside the convex hull of the support of the operator’s spectral measure). In the present paper, we study whether the obtained perturbation of the original operator is compact. For limit-periodic Jacobi operators, we obtain a necessary and sufficient condition on the location of the masses for the perturbation to be compact.
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References
N. I. Akhiezer, The Classical Moment Problem and Some Related Topics in Analysis (Fizmatgiz, Moscow, 1961) [in Russian].
E.M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality (Nauka, Moscow, 1988) [in Russian].
T. Kato, Perturbation Theory for Linear Operators, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1966), Vol. 132.
A. A. Gonchar, “The convergence of Padé approximants for certain classes of meromorphic functions,” Mat. Sb. 97(4), 607–629 (1975) [Math. USSR-Sb. 26, 555–575 (1975)].
E. M. Nikishin, “Discrete Sturm-Liouville operator and some problems of function theory,” in Proc. I. G. Petrovskii Seminar (Moscow, Izd.Moskov. Univ., 1984), Vol. 10, pp. 3–77 [in Russian].
F. Marcellán and P. Maroni, “Sur l’adjoinction d’unemasse de Dirac à une forme régulière et semi-classique,” Ann. Mat. Pura Appl. (4) 162(1), 1–22 (1992).
D. Damanik, R. Killip, and B. Simon, Perturbations of Orthogonal Polynomials with Periodic Recursion Coefficients, arXiv: math. SP/0702388v2.
E. A. Rakhmanov, “On the asymptotics of the ratio of orthogonal polynomials. II,” Mat. Sb. 118(1), 104–117 (1982) [Math. USSR-Sb. 46 (3), 105–117 (1983)].
S. P. Suetin, “Trace formulas for a class of Jacobi operators,” Mat. Sb. 198(6), 107–138 (2007) [Russian Acad. Sci. Sb. Math. 198 (6), 857–885 (2007)].
V. Batchenko and F. Gesztesy, “On the spectrum of Jacobi operators with quasiperiodic algebro-geometric coefficients,” IMRP Int. Math. Res. Pap. 10, 511–563 (2005).
B. Beckermann, “Complex Jacobi matrices,” J. Comput. Appl. Math. 127(1–2), 17–65 (2001).
H. Widom, “Extremal polynomials associated with a system of curves in the complex plane,” Adv. in Math. 3(2), 127–232 (1969).
A. I. Aptekarev, “Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices. Mat. Sb. 125(2), 231–258 (1984) [Math. USSR-Sb. 53, 233–260 (1986)].
V. A. Kalyagin and A. A. Kononova, “On asymptotics of polynomials orthogonal on an arc system with respect to a measure with a discrete part,” Algebra i Analyz 21(4), 71–91 (2009) [St. Petersburg Math. J.].
A. A. Gonchar and S. P. Suetin, “On Padé approximants of meromorphic functions of Markov type,” in Current Problems in Mathematics (MIAN, Moscow, 2004), Vol. 5, pp. 3–67 [in Russian].
E. A. Rakhmanov, “Convergence of diagonal Padé approximants,” Mat. Sb. 104(2), 271–291 (1977) [Math. USSR-Sb. 33, 243–260 (1977)].
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Original Russian Text © V. A. Kalyagin, A. A. Kononova, 2009, published in Matematicheskie Zametki, 2009, Vol. 86, No. 6, pp. 845–858.
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Kalyagin, V.A., Kononova, A.A. On compact perturbations of the limit-periodic Jacobi operator. Math Notes 86, 789–800 (2009). https://doi.org/10.1134/S0001434609110212
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DOI: https://doi.org/10.1134/S0001434609110212