On rank-one perturbations of normal operators, II

AS the title indicates, this note is a sequel to [2], in which we showed that a large class of rank-one perturbations of a diagonalizable normal operator have nontrivial hyperinvariant subspaces. Below we establish the perhaps surprising fact that the commutants of such operators are abelian, paralleling thereby the properties of the commutants of normal operators of multiplicity one. We also show by example that this behavior does not extend to the commutants of rank-one perturbations of all normal operators of multiplicity one, and we discuss similarity and quasisimilarity questions associated with this class of operators.