Spectral decomposability of rank-one perturbations of normal operators

This paper is a continuation of the study by Foias, Jung, Ko, and Pearcy (2007) [4], and Foias, Jung, Ko, and Pearcy (2008) [5], of rank-one perturbations of diagonalizable normal operators. In Foias, Jung, Ko, and Pearcy (2007) [4], we showed that there is a large class of such operators each of which has a nontrivial hyperinvariant subspace, and in Foias, Jung, Ko, and Pearcy (2008) [5], we proved that the commutant of each of these rank-one perturbations is abelian. In this paper we show that the operators considered in Foias, Jung, Ko, and Pearcy (2007) [4], have more structure – namely, that they are decomposable operators in the sense of Colojoară and Foias (1968) [1].