Discrete spectra for some complex infinite band matrices

Abstract

Under suitable assumptions the eigenvalues for an unbounded discrete operator $A$ in $l_2$, given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let $\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},$ where ${\rm Lim}_{n\to \infty} \lambda_n$ is the set of all limit points of the sequence $(\lambda_{n})$ and $A_n$ is a finite dimensional orthogonal truncation of $A$. The aim of this article is to provide the conditions that are sufficient for the relations $\sigma(A) \subset \Lambda(A)$ or $\Lambda (A) \subset \sigma (A)$ to be satisfied for the band operator $A$.