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

This paper is concerned with the union spΩ(j,k) Tn(a) of all possible spectra that may emerge when perturbing a large n × n Toeplitz band matrix Tn(a) in the (j, k) site by a number randomly chosen from some set Ω. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of spΩ(j,k) Tn(a) as n → ∞. Also discussed are the cases of small and large sets Ω as well as the "discontinuity of the infinite volume case", which means that in general spΩ(j,k) Tn(a) does not converge to something close to spΩ(j,k) Tn(a) as n → ∞ where T(a) is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case.

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