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The spectra of large Toeplitz band matrices with a randomly perturbed entry


Authors: A. Böttcher, M. Embree and V. I. Sokolov
Journal: Math. Comp. 72 (2003), 1329-1348
MSC (2000): Primary 47B35, 65F15; Secondary 15A18, 47B80, 82B44
DOI: https://doi.org/10.1090/S0025-5718-03-01505-9
Published electronically: February 3, 2003
MathSciNet review: 1972739
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Abstract: This paper is concerned with the union $\operatorname{sp}_\Omega^{(j,k)} T_n(a)$of all possible spectra that may emerge when perturbing a large $n \times n$ Toeplitz band matrix $T_n(a)$ in the $(j,k)$ site by a number randomly chosen from some set $\Omega$. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of $\operatorname{sp}_\Omega^{(j,k)} T_n(a)$ as $n \to \infty$. Also discussed are the cases of small and large sets $\Omega$ as well as the ``discontinuity of the infinite volume case'', which means that in general $\operatorname{sp}_\Omega^{(j,k)} T_n(a)$ does not converge to something close to $\operatorname{sp}_\Omega^{(j,k)} T(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case.


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Additional Information

A. Böttcher
Affiliation: Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany
Email: aboettch@mathematik.tu-chemnitz.de

M. Embree
Affiliation: Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom
Address at time of publication: Department of Computational and Applied Mathematics, Rice University, 6100 Main Street – MS 134, Houston, Texas 77005–1892
Email: embree@rice.edu

V. I. Sokolov
Affiliation: Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany
Address at time of publication: Institut für Mathematik, TU Berlin, 10623 Berlin, Germany
Email: sokolov@math.tu-berlin.de

DOI: https://doi.org/10.1090/S0025-5718-03-01505-9
Keywords: Toeplitz operator, pseudospectrum, random perturbation
Received by editor(s): August 3, 2001
Published electronically: February 3, 2003
Additional Notes: The work of the second author was supported by UK Engineering and Physical Sciences Research Council Grant GR/M12414.
Article copyright: © Copyright 2003 American Mathematical Society

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