Absence of the point spectrum in a class of tridiagonal operators

Abstract

Let T be the tridiagonal operator Te n= c n e n+1+ c n−1 e n−1+b ne n , Te 1= c 1 e 2+b 1e 1 , where c n and b n are real sequences with c n >0, lim n→∞ c n = c, lim n→∞ b n = b, n=1,2,…, and e n is an orthonormal basis in a Hilbert space H. The spectrum of T consists of the interval [−2 c +b,2 c +b] plus a point spectrum outside the interval, which may be empty, finite or denumerable with accumulation points the points −2 c +b or 2 c +b . Here sufficient conditions are given such that the point spectrum of the operator T outside the interval [−2 c +b,2 c +b] is empty, which means that the spectrum of T is the entire interval [−2 c +b,2 c +b] . The results are illustrated with examples.