Infinite Toeplitz and Laurent matrices with localized impurities

This paper is concerned with the change of the spectra of infinite Toeplitz and Laurent matrices under perturbations in a prescribed finite set of sites. The main result says that the spectrum of a Toeplitz matrix with a non-constant rational symbol is not affected by small localized impurities, while such impurities can nevertheless enlarge the spectrum of the corresponding Laurent matrix. We also study the spectra that may emerge when randomly perturbing Toeplitz or Laurent matrices in a randomly chosen single site.

[1]  Spectral properties of random non-self-adjoint matrices and operators , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  David R. Nelson,et al.  Vortex pinning and non-Hermitian quantum mechanics , 1997 .

[3]  Richard M. Beam,et al.  The Asymptotic Spectra of Banded Toeplitz and Quasi-Toeplitz Matrices , 1993, SIAM J. Sci. Comput..

[4]  Mark Ainsworth,et al.  The graduate student's guide to numerical analysis '98 : lecture notes from the VIII EPSRC Summer School in Numerical Analysis , 1999 .

[5]  L. Trefethen,et al.  Eigenvalues and pseudo-eigenvalues of Toeplitz matrices , 1992 .

[6]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[7]  Non-Hermitian localization and delocalization. , 1997, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Diederich Hinrichsen,et al.  Spectral value sets: a graphical tool for robustness analysis , 1993 .

[9]  D. Hinrichsen,et al.  Real and Complex Stability Radii: A Survey , 1990 .

[10]  E. Gallestey,et al.  Spectral value sets of closed linear operators , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[12]  A. Böttcher.  Pseudospectra and Singular Values of Large Convolution Operators , 1994 .

[13]  L. Trefethen,et al.  Spectra, pseudospectra, and localization for random bidiagonal matrices , 2000, cond-mat/0003514.

[14]  V. I. Sokolov,et al.  The spectra of large Toeplitz band matrices with a randomly perturbed entry , 2003, Math. Comput..

[15]  L. A. Coburn,et al.  Weyl's theorem for nonnormal operators , 1966 .

[16]  H. Landau,et al.  On Szegö’s eingenvalue distribution theorem and non-Hermitian kernels , 1975 .