Almost-invariant half-spaces of operators on $ω$

Abstract

We study the Almost-Invariant Subspace Problem for operators defined on non-normable Fréchet spaces and especially on $ω$. 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 $ω$ by showing that an operator $T$ on $ω$ possesses a closed almost-invariant half-space if and only if $T$ is conjugate to $F + R + λ\operatorname{Id}$ or to $B + R + λ\operatorname{Id}$ where $F$ is the Forward shift and $B$ is the Backward shift and where $R$ is a finite rank operator and $λ\in \mathbb{C}$. We also investigate the case of operators defined on $X \oplus ω$ where $X$ is an infinite-dimensional Fréchet space with a continuous norm and we show that in this case, every operator defined on $X \oplus ω$ possesses a closed almost-invariant half-space.