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

Adi Tcaciuc

Published 2020-06-22Version 1

We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, any for any $\varepsilon>0$, there exists an operator $F$ of rank at most one 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.