The invariant subspace problem for rank one perturbations

TL;DR

This paper demonstrates that any bounded operator on an infinite-dimensional Banach space can be minimally perturbed to acquire an infinite-dimensional invariant subspace, advancing the understanding of the invariant subspace problem.

Contribution

It establishes the existence of rank-one perturbations that induce invariant subspaces, even under minimal spectral conditions, extending previous results in operator theory.

Findings

Existence of rank-one perturbations creating invariant subspaces

Small norm perturbations are sufficient under certain spectral conditions

Finite rank perturbations can be used when spectral conditions are not met

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 $T^*$ 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.