Absence of the point spectrum in a class of tridiagonal operators

摘要

Let T be the tridiagonal operator Ten=cnen+1+cn−1en−1+bnen, Te1=c1e2+b1e1, where cn and bn are real sequences with cn>0, limn→∞cn=c, limn→∞bn=b, n=1,2,…, and en is an orthonormal basis in a Hilbert space H. The spectrum of T consists of the interval [−2c+b,2c+b] plus a point spectrum outside the interval, which may be empty, finite or denumerable with accumulation points the points −2c+b or 2c+b. Here sufficient conditions are given such that the point spectrum of the operator T outside the interval [−2c+b,2c+b] is empty, which means that the spectrum of T is the entire interval [−2c+b,2c+b]. The results are illustrated with examples.