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

X iv :m at h/ 05 12 22 2v 1 [ m at h. SP ] 1 1 D ec 2 00 5 The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences Leonid Golinskii and Stefano Serra-Capizzano 1 Abstract Under the mild trace-norm assumptions, we show that the eigenvalues of a generic (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 matrixvalued symbols, which arises when dealing with orthogonal polynomials with asymptotically periodic recurrence coefficients.