Invariant subspace problem for rank-one perturbations: the quantitative version (Q2672207)

Let \(X\) be an infinite dimensional complex Banach space and \((\mathcal{B}(X),\|\cdot\|)\) be the set of bounded operators endowed with the norm \(\|\cdot\|\). The purpose of the author is to show that for any \(\varepsilon>0\), there is \(F\in\mathcal{B}(X)\) such that \(T+F\) has an invariant half-space where \(T\in\mathcal{B}(X)\).