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.
@article{bwmeta1.element.doi-10_1515_conop-2016-0010,
author = {Andrey Osipov},
title = {A study of resolvent set for a class of band operators with matrix elements},
journal = {Concrete Operators},
volume = {3},
year = {2016},
pages = {85-93},
zbl = {06587143},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_conop-2016-0010}
}
Andrey Osipov. A study of resolvent set for a class of band operators with matrix elements. Concrete Operators, Tome 3 (2016) pp. 85-93. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_conop-2016-0010/
[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. | Zbl 0824.47017
[2] B. Beckermann, On the classification of the spectrum of second order difference operators, Mathematische Nachrichten, 2000, 216, 45-59. | Zbl 0969.47027
[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. | Zbl 0897.39003
[4] B. Beckermamm, A. Osipov, Some spectral properties of infinite band matrices, Numerical Algorithms, 2003, 34, 173-185.
[5] J. M. Berezanskij, Expansions in eigenfunctions of selfadjoint operators, A.M.S. Providence, R. I. 1968.
[6] M. M. Gekhtman, Integration of non-Abelian Toda-type chains. Functional Analysis and its Applications, 1990, 24, no. 3, 231-233.
[7] S. Demko, W. F. Moss, P. W. Smith, Decay Rates for Inverses of Band Matrices, Math. Comp., 1984, 43, 491-499. | Zbl 0568.15003
[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.
[9] A. Osipov, Some properties of resolvent sets of second order difference operators with matrix coefficients. Mathematical Notes, 2000, 68, no. 6, 806-809. | Zbl 0993.39012
[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. | Zbl 1033.47017
[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. | Zbl 0984.37023