标题： 关于线性动力学中的空间性 显示英文标题

标题： On Spaceability within Linear Dynamics

摘要： 我们从结构角度研究线性动力学中的空间可容纳现象。 给定一个连续线性算子\(T:X \to X\)，我们引入集合\(\Omega(T)\)，该集合由所有连续线性算子\(h:X \to X\)组成，其中存在一个严格递增的正整数序列\((\theta_n)_n\)，使得集合\(\{x \in X : \displaystyle{\lim_{n \rightarrow \infty} T^{\theta_n}x = h(x)}\}\)在\(X\)中是稠密的。 在此框架下，两个经典现象——在可分无限维复巴拿赫空间中存在超循环和循环子空间——作为由\(\Omega(T)\)描述的共同底层结构的实例浮现出来。 为了分析\(\Omega(T)\)，我们引入了同时被\(T\)近似（c.s.a.）的集合的概念，并证明每个最大的c.s.a.是一个SOT闭的仿射流形。对于可分巴拿赫空间上的准刚性算子，我们建立了包含恒等算子的唯一最大c.s.a.的存在性。此外，我们通过左乘算子\(L_T\)在有界算子代数上的作用来研究\(\Omega(T)\)。我们的方法结合了两个关键要素：对准刚性算子的递归子空间的A. López技术的改进，以及由第一作者和A. Arbieto获得的共同稠密线性化结果。这些工具为集合\(\Omega(T)\)、\(\mathcal{AP}\Omega(T)\)以及任何可数的c.s.a. by\(T\)提供了新的空间性结果。 显示英文摘要

摘要： We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator \(T:X \to X\), we introduce the set \(\Omega(T)\), consisting of all continuous linear operators \(h:X \to X\) for which there exists a strictly increasing sequence \((\theta_n)_n\) of positive integers such that the set \(\{x \in X : \displaystyle{\lim_{n \rightarrow \infty} T^{\theta_n}x = h(x)}\}\) is dense in \(X\). Within this framework, two classical phenomena--the existence of hypercyclic and recurrent subspaces in separable infinite-dimensional complex Banach spaces--emerge as instances of a common underlying structure described by \(\Omega(T)\). To analyze \(\Omega(T)\), we introduce the notion of collections simultaneously approximated (c.s.a.) by \(T\), and show that every maximal c.s.a. is an SOT-closed affine manifold. For quasi-rigid operators on separable Banach spaces, we establish the existence of a unique maximal c.s.a. containing the identity operator. Furthermore, we examine \(\Omega(T)\) through the left-multiplication operator \(L_T\) acting on the algebra of bounded operators. Our approach combines two key ingredients: a refinement of A. L\'opez's technique on recurrent subspaces for quasi-rigid operators, and a common dense-lineability result obtained by the first author and A. Arbieto. These tools yield new spaceability results for the sets \(\Omega(T)\), \(\mathcal{AP}\Omega(T)\), and for any countable c.s.a. by \(T\).