The asymptotic properties of the spectrum of non symmetrically perturbed   Jacobi matrix sequences
  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 $[0,\pi]$, which characterizes the nonperturbed case. In this way the real interval $[-2,2]$ is still a cluster for the asymptotic joint spectrum and, moreover, $[-2,2]$ 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 symbol, which arises when dealing with orthogonal polynomials with asymptotically periodic recurrence coefficients. 
Leonid Golinskii and Stefano Serra-Capizzano