Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 17, 2016

A study of resolvent set for a class of band operators with matrix elements

  • Andrey Osipov
From the journal Concrete Operators


For operators generated by a certain class of infinite three-diagonal matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying second order finite-difference equations. This enables us to describe some asymptotic behavior of the corresponding systems of vector orthogonal polynomials on the resolvent set. We also find that the operators generated by infinite Jacobi matrices have the largest resolvent set in this class.


[1] A. Aptekarev, V. Kaliaguine and W. Van Assche, Criterion for the resolvent set of nonsymmetric tridiagonal matrix, Proc. Amer. Math. Soc., 1995, vol 123, no. 8, 2423-2430. 10.1090/S0002-9939-1995-1254830-5Search in Google Scholar

[2] B. Beckermann, On the classification of the spectrum of second order difference operators, Mathematische Nachrichten, 2000, 216, 45-59. 10.1002/1522-2616(200008)216:1<45::AID-MANA45>3.0.CO;2-RSearch in Google Scholar

[3] B. Beckermann and V. A. Kaliaguine, The diagonal of the Padé table and the approximation of the Weyl function of second order difference operators, Constructive Approximation, 1997, 13, 481-510. 10.1007/s003659900056Search in Google Scholar

[4] B. Beckermamm, A. Osipov, Some spectral properties of infinite band matrices, Numerical Algorithms, 2003, 34, 173-185. 10.1023/B:NUMA.0000005361.17723.a4Search in Google Scholar

[5] J. M. Berezanskij, Expansions in eigenfunctions of selfadjoint operators, A.M.S. Providence, R. I. 1968. 10.1090/mmono/017Search in Google Scholar

[6] M. M. Gekhtman, Integration of non-Abelian Toda-type chains. Functional Analysis and its Applications, 1990, 24, no. 3, 231-233. 10.1007/BF01077968Search in Google Scholar

[7] S. Demko, W. F. Moss, P. W. Smith, Decay Rates for Inverses of Band Matrices, Math. Comp., 1984, 43, 491-499. 10.1090/S0025-5718-1984-0758197-9Search in Google Scholar

[8] V. A. Kaliaguine, Hermite-Padé approximants and spectral analysis of nonsymmetric operators, Mat. Sb., 1994, 185, 79-100 (In Russian). English transl. in Russian Acad. Sci. Sb. Math., 1995, 82, 199-216. 10.1070/SM1995v082n01ABEH003558Search in Google Scholar

[9] A. Osipov, Some properties of resolvent sets of second order difference operators with matrix coefficients. Mathematical Notes, 2000, 68, no. 6, 806-809. 10.1023/A:1026681204898Search in Google Scholar

[10] A. Osipov, On some issues related to the moment problem for the band matrices with operator elements, Journal of Mathematical Analysis and Applications, 2002, 275, no. 2, 657-675. 10.1016/S0022-247X(02)00372-4Search in Google Scholar

[11] V. Sorokin, J. Van Iseghem, Matrix Hermite-Pade problem and dynamical systems. Journ. of Computational and Applied Mathematics, 2000, 122, no. 1-2, 275-295. 10.1016/S0377-0427(00)00354-XSearch in Google Scholar

Received: 2015-12-9
Accepted: 2016-5-3
Published Online: 2016-5-17

© 2016 Andrey Osipov

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 1.4.2023 from
Scroll to top button