Limit points of eigenvalues of truncated tridiagonal operators

Abstract

Let T be the tridiagonal operator Te n = a n e n+1 + a n−1 e n−1 + b n e n , Te 1= a 1 e 2+ b 1 e 1, acting on a fixed orthonormal basis { e n }, n=1,2,…, of a Hilbert space H. Let P N be the orthogonal projection on the finite-dimensional space H N spanned by the elements { e 1, e 2,…, e N } and let T N be the truncated operator T N = P N TP N . If T has a unique self-adjoint extension then the set Λ(T)={λ : there exists a sequence of eigenvalues λ N of T N with the property λ N→λ} contains the spectrum σ( T) of T and examples show that, in general, σ( T)≠ Λ( T). For many reasons, the knowledge of the equality σ( T)= Λ( T) is important. In this paper sufficient conditions are presented such that σ( T)= Λ( T).