The invariant subspace problem for rank-one perturbations

Abstract

We show that for any bounded operator $T$ acting on an infinite-dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $T$ or $T^{\ast }$ 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.