Invariant subspace problem for rank-one perturbations: the quantitative version

TL;DR

This paper proves that for any bounded operator on an infinite-dimensional complex Banach space, a small rank-one perturbation can create an operator with an infinite-dimensional invariant subspace, solving a key aspect of the invariant subspace problem.

Contribution

It establishes the full generality of the quantitative invariant subspace problem for rank-one perturbations, removing previous spectral restrictions.

Findings

Existence of rank-one perturbations producing invariant subspaces

Applicable to any bounded operator on infinite-dimensional spaces

Advances understanding of the invariant subspace problem

Abstract

We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $\varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $\varepsilon$ such that $T+F$ has an invariant subspace of infinite dimension and codimension. A version of this result was proved in \cite{T19} under additional spectral conditions for $T$ or $T^*$. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.