Tridiagonal matrix representations of cyclic selfadjoint operators

Abstract

A bounded cyclic self-adjoint operator C, defined on a separable Hubert space, can be represented as a tridiagonal matrix with respect to the basis generated by a cyclic vector. If the main diagonal entries are zeros, C may be regarded as the real part of a weighted shift operator. Define / to be the corresponding imaginary part and it follows that CJ -JC = -2iK where K is a diagonal operator. The main purpose of this paper is to show that if the subdiagonal entries converge to a non-zero limit and if K is of trace class then C has an absolutely continuous part.