Similarity of perturbations of Hessenberg matrices

To every infinite lower Hessenberg matrix D is associated a linear operator on l2. In this paper we prove the similarity of the operator D − ∆, where ∆ belongs to a certain class of compact operators, to the operator D−∆′, where ∆′ is of rank one. We first consider the case when ∆ is lower triangular and has finite rank; then we extend this to ∆ of infinite rank assuming that D is bounded. In Section 3 we examine the cases when D = S and D = (S + S)/2, where S denotes the unilateral shift.