Finite rank perturbations of normal operators: spectral subspaces and Borel series

We characterize the spectral subspaces associated to closed sets of rank-one perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space by means of functional equations involving Borel series. As a particular instance, if T=D¿+u ¿vis a rank-one perturbation of a diagonalizable normal operator D¿with respect to a basis E=(en)n¿1and the vectors uand vhave Fourier coefficients (¿n)n¿1and (ßn)n¿1with respect to E, respectively, it is shown that Thas non-trivial closed invariant subspaces provided that either (¿n)n¿1¿1or (ßn)n¿1¿1. Likewise, analogous results hold for finite rank perturbations of D¿. Moreover, such operators Thave non-trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity extending previous theorems of Foia¿, Jung, Ko and Pearcy [8]and of Fang and J. Xia [6]on an open question of at least forty years.

rank-one perturbation of normal operators; rank-one perturbation of diagonal operators; spectral subspaces; borel series; wolff-denjoy series

Finite rank perturbations of normal operators: spectral subspaces and Borel series Articles