We show that a bounded quasinilpotent operator T acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator F and a scalar α∈ℂ, α≠0, α≠1, such that T+F and T+αF are also quasinilpotent. We also prove that for any fixed rank-one operator F, almost all perturbations T+αF have invariant subspaces of infinite dimension and codimension.
