The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences

Details The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences

2005-12-11

arxiv.org/abs/math/0512222v1

Mathematics

Spectral Theory

LaTeX, 19 pages

Scientific paper

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.

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