Finite rank perturbations of normal operators: Spectral idempotents and decomposability

Abstract

We prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5–7], Fang and J. Xia [3] and the authors [8,9] on an open question posed by Pearcy in the seventies.