A bounded cyclic self-adjoint
operator C, defined on a separable Hilbert 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 J 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.
