The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences

Abstract

Under the mild trace-norm assumptions, we show that the eigenvalues of an arbitrary (non-Hermitian) complex perturbation of a Jacobi matrix sequence (not necessarily real) are still distributed as the real-valued function 2 cos t on [ 0 , π ] which characterizes the nonperturbed case. In this way the real interval [ - 2 , 2 ] is still a cluster for the asymptotic joint spectrum and, moreover, [ - 2 , 2 ] still attracts strongly (with infinite order) the perturbed matrix sequence. The results follow in a straightforward way from more general facts that we prove in an asymptotic linear algebra framework and are plainly generalized to the case of matrix-valued symbols, which arises when dealing with orthogonal polynomials with asymptotically periodic recurrence coefficients.