  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 $\alpha\in\mathbb{C}$, $\alpha\neq 0$,
$\alpha\neq 1$, such that $T+F$ and $T+\alpha F$ are also quasinilpotent. We
also prove that for any fixed rank-one operator $F$, almost all perturbations
$T+\alpha F$ have invariant subspaces of infinite dimension and codimension.
