Suppose A is a compact normal operator on a Hilbert space H with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let L be its rank one perturbation. We show that either infinitely many moment equalities hold or the linear span of root vectors of L, corresponding to non-zero eigenvalues, is of finite codimension in H. In contrast to classical results, we do not assume the perturbation to be weak.