Almost-invariant and essentially-invariant halfspaces

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if $X$ is an infinite-dimensional complex Banach space then every operator $T\in \mathcal{L}(X)$ admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well.