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.
2007
GLT sequence; Jacobi matrix; Joint eigenvalue distribution; Matrix sequence; Mergelyan Theorem
Golinskii, L.; SERRA CAPIZZANO, Stefano
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/1671172
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 53
  • ???jsp.display-item.citation.isi??? 50
social impact