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

Abstract

We show that for any bounded operator T acting on an infinite dimensional complex Banach space, and for any ε>0, there exists an operator F of rank at most one and norm smaller than ε such that T+F has an invariant subspace of infinite dimension and codimension. A version of this result was proved in [15] 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.