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.
The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences
SERRA CAPIZZANO, STEFANO
2007-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.