Reducing subspaces for rank-one perturbations of normal operators

We study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator T = D + u circle times v (uniquely determined by such expression) is contained in a line, since in such a case T has a reducing subspace if and only if T is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators T = D + u circle times v with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of T is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces M of T such that the restriction T vertical bar(M) is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wenner) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.