On quasinilpotent operators and the invariant subspace problem
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.
Tcaciuc, A. (2019). On quasinilpotent operators and the invariant subspace problem. Journal of Mathematical Analysis and Applications 477(1), 187-195. https://doi.org/10.1016/j.jmaa.2019.04.026
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