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

Abstract

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=DΛ+u⊗v is a rank-one perturbation of a diagonalizable normal operator DΛ with respect to a basis E=(en)n≥1 and the vectors u and v have Fourier coefficients (αn)n≥1 and (βn)n≥1 with respect to E, respectively, it is shown that T has non-trivial closed invariant subspaces provided that either (αn)n≥1∈ℓ1 or (βn)n≥1∈ℓ1. Likewise, analogous results hold for finite rank perturbations of DΛ. Moreover, such operators T have non-trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity extending previous theorems of Foiaş, Jung, Ko and Pearcy [8] and of Fang and J. Xia [6] on an open question of at least forty years.