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

Abstract

We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $\epsilon >0$, there exists an operator $F$ of rank at most one and norm smaller than $\epsilon$ 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 $T^{\ast }$. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.