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

Andrey Osipov

Concrete Operators (2016)

  • Volume: 3, Issue: 1, page 85-93
  • ISSN: 2299-3282

Abstract

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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.

How to cite

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Andrey Osipov. "A study of resolvent set for a class of band operators with matrix elements." Concrete Operators 3.1 (2016): 85-93. <http://eudml.org/doc/277082>.

@article{AndreyOsipov2016,
abstract = {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.},
author = {Andrey Osipov},
journal = {Concrete Operators},
keywords = {Band operators; Difference; Equations; Weyl matrix, Orthogonal polynomials, Resolvent sets; band operators; difference; equations; Weyl matrix; orthogonal polynomials; resolvent sets},
language = {eng},
number = {1},
pages = {85-93},
title = {A study of resolvent set for a class of band operators with matrix elements},
url = {http://eudml.org/doc/277082},
volume = {3},
year = {2016},
}

TY - JOUR
AU - Andrey Osipov
TI - A study of resolvent set for a class of band operators with matrix elements
JO - Concrete Operators
PY - 2016
VL - 3
IS - 1
SP - 85
EP - 93
AB - 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.
LA - eng
KW - Band operators; Difference; Equations; Weyl matrix, Orthogonal polynomials, Resolvent sets; band operators; difference; equations; Weyl matrix; orthogonal polynomials; resolvent sets
UR - http://eudml.org/doc/277082
ER -

References

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  1. [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.  Zbl0824.47017
  2. [2] B. Beckermann, On the classification of the spectrum of second order difference operators, Mathematische Nachrichten, 2000, 216, 45-59.  Zbl0969.47027
  3. [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.  Zbl0897.39003
  4. [4] B. Beckermamm, A. Osipov, Some spectral properties of infinite band matrices, Numerical Algorithms, 2003, 34, 173-185.  
  5. [5] J. M. Berezanskij, Expansions in eigenfunctions of selfadjoint operators, A.M.S. Providence, R. I. 1968.  
  6. [6] M. M. Gekhtman, Integration of non-Abelian Toda-type chains. Functional Analysis and its Applications, 1990, 24, no. 3, 231-233.  
  7. [7] S. Demko, W. F. Moss, P. W. Smith, Decay Rates for Inverses of Band Matrices, Math. Comp., 1984, 43, 491-499.  Zbl0568.15003
  8. [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. [9] A. Osipov, Some properties of resolvent sets of second order difference operators with matrix coefficients. Mathematical Notes, 2000, 68, no. 6, 806-809.  Zbl0993.39012
  10. [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.  Zbl1033.47017
  11. [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.  Zbl0984.37023

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