We show that for any bounded operator acting on an infinite-dimensional Banach space there exists an operator of rank at most one such that has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of or does not consist entirely of eigenvalues, we can find such rank-one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite-rank perturbations of arbitrarily small norm, but not necessarily of rank one.
"The invariant subspace problem for rank-one perturbations." Duke Math. J. 168 (8) 1539 - 1550, 1 June 2019. https://doi.org/10.1215/00127094-2018-0071