  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.
