Abstract:Given a Polish topology $\tau$ on $\mathcal {B}_{1}(X)$, the set of all contraction operators on $X=\ell _p$, $1\le p<\infty$ or $X=c_0$, we prove several results related to the following question: does a typical $T\in \mathcal {B}_{1}(X)$ in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense $G_\delta$ set $\mathcal G\subseteq (\mathcal {B}_{1}(X),\tau )$ such that every $T\in \mathcal G$ has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong$^*$ Operator Topology.

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中文翻译：

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典型的ℓ_{𝑝}-空间收缩是否具有非平凡的不变子空间？

摘要：给定 $\mathcal {B}_{1}(X)$ 上的波兰拓扑 $\tau$，$X=\ell _p$、$1\le p<\infty$ 或$X=c_0$，我们证明了与以下问题相关的几个结果：Baire Category 意义上的典型 $T\in \mathcal {B}_{1}(X)$ 是否具有非平凡的不变子空间？换句话说，是否有一个密集的 $G_\delta$ 集 $\mathcal G\subseteq (\mathcal {B}_{1}(X),\tau )$ 使得每个 $T\in \mathcal G$ 有一个非平凡的不变子空间？我们主要关注强运算符拓扑和 Strong$^*$ 运算符拓扑。

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