Invariant half-spaces for rank-one perturbations

TL;DR

This paper proves that for any bounded linear operator on an infinite-dimensional Banach space, a small rank-one perturbation can create an operator with an infinite-dimensional invariant subspace, improving previous results.

Contribution

It introduces a method to find small rank-one perturbations that guarantee the existence of invariant subspaces for bounded operators.

Findings

Existence of rank-one perturbations with arbitrarily small norm

Creation of infinite-dimensional invariant subspaces via perturbation

Improvement over previous invariant subspace results

Abstract

If T is a bounded linear operator acting on an infinite-dimensional Banach space, then there exists and operator F of rank at most one and arbitrarily small norm such that T-F has an invariant subspace of infinite dimension and codimension. This improves results of Tcaciuc and other authors.