Almost-invariant half-spaces of operators on $\omega$

We study the Almost-Invariant Subspace Problem for operators defined on non-normable Fréchet spaces and especially on $\omega$. This problem is stated as follows: “Given a bounded operator $T$ on an infinite-dimensional complex Fréchet space $X$, can we always find a closed almost-invariant half-space?”. In this paper we solve the problem for the space $\omega$ by showing that an operator $T$ on $\omega$ possesses a closed almost-invariant half-space if and only if $T$ is conjugate to $F+R+\lambda \mathrm{Id}$ or to $B+R+\lambda \mathrm{Id}$ where $F$ is the Forward shift and $B$ is the Backward shift and where $R$ is a finite rank operator and $\lambda \in \mathbb{C}$. We also investigate the case of operators defined on $X\oplus \omega$ where $X$ is an infinite-dimensional Fréchet space with a continuous norm and we show that in this case, every operator defined on $X\oplus \omega$ possesses a closed almost-invariant half-space.