Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.06156 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912575730483200 |
|---|---|
| author | Saavedra, Manuel Stadlbauer, Manuel |
| author_facet | Saavedra, Manuel Stadlbauer, Manuel |
| contents | We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator \(T:X \to X\), we introduce the set \(Ω(T)\), consisting of all continuous linear operators \(h:X \to X\) for which there exists a strictly increasing sequence \((θ_n)_n\) of positive integers such that the set \(\{x \in X : \displaystyle{\lim_{n \rightarrow \infty} T^{θ_n}x = h(x)}\}\) 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 \(Ω(T)\). To analyze \(Ω(T)\), 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 \(Ω(T)\) through the left-multiplication operator \(L_T\) acting on the algebra of bounded operators. Our approach combines two key ingredients: a refinement of A. López'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 \(Ω(T)\), \(\mathcal{AP}Ω(T)\), and for any countable c.s.a. by \(T\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06156 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Spaceability within Linear Dynamics Saavedra, Manuel Stadlbauer, Manuel Functional Analysis We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator \(T:X \to X\), we introduce the set \(Ω(T)\), consisting of all continuous linear operators \(h:X \to X\) for which there exists a strictly increasing sequence \((θ_n)_n\) of positive integers such that the set \(\{x \in X : \displaystyle{\lim_{n \rightarrow \infty} T^{θ_n}x = h(x)}\}\) 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 \(Ω(T)\). To analyze \(Ω(T)\), 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 \(Ω(T)\) through the left-multiplication operator \(L_T\) acting on the algebra of bounded operators. Our approach combines two key ingredients: a refinement of A. López'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 \(Ω(T)\), \(\mathcal{AP}Ω(T)\), and for any countable c.s.a. by \(T\). |
| title | On Spaceability within Linear Dynamics |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2509.06156 |