Finite rank perturbations of normal operators: spectral idempotents and decomposability

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 Colojoar¿ and Foia¿ [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 Foia¿, Jung, Ko and Pearcy [5], [6], [7], Fang and J. Xia [3] and the authors [8], [9] on an open question posed by Pearcy in the seventies.

finite rank perturbations of normal operators; invariant subspaces; spectral subspaces; decomposable operators

Finite rank perturbations of normal operators: spectral idempotents and decomposability Articles