发表机构
Université de Mons（蒙斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了ω空间上算子的几乎不变子空间问题，明确了该类算子具有闭几乎不变半空间的充要条件，还证明了X⊕ω上的算子均具有闭几乎不变半空间。
AI 中文摘要
我们研究定义在不可赋范Fréchet空间、尤其是ω上算子的几乎不变子空间问题，该问题表述为：给定无限维复Fréchet空间X上的有界算子T，是否总能找到闭的几乎不变半空间？本文解决了ω空间的该问题，证明ω上算子T具有闭几乎不变半空间当且仅当T共轭于F + R + λId或B + R + λId，其中F为前向移位、B为后向移位、R为有限秩算子，λ∈ℂ。我们还研究了X⊕ω上算子的情况，X为具有连续范数的无限维Fréchet空间，证明该空间上的每个算子都具有闭几乎不变半空间。
英文摘要
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.
Comments23 pages
