Recurrent subspaces in Banach spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914829658226688 |
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| author | López-Martínez, Antoni |
| author_facet | López-Martínez, Antoni |
| contents | We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular: we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace; when $X$ is a complex Banach space we show that having a recurrent subspace is equivalent to the fact that the essential spectrum of the operator intersects the closed unit disc; and we extend the previous result to the real case. As a consequence we obtain that: a weakly-mixing operator on a real or complex separable Banach space has a hypercyclic subspace if and only if it has a recurrent subspace. The results exposed exhibit a symmetry between the hypercyclic and recurrence spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_04464 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Recurrent subspaces in Banach spaces López-Martínez, Antoni Functional Analysis Dynamical Systems 47A16, 47A53, 37B20 We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular: we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace; when $X$ is a complex Banach space we show that having a recurrent subspace is equivalent to the fact that the essential spectrum of the operator intersects the closed unit disc; and we extend the previous result to the real case. As a consequence we obtain that: a weakly-mixing operator on a real or complex separable Banach space has a hypercyclic subspace if and only if it has a recurrent subspace. The results exposed exhibit a symmetry between the hypercyclic and recurrence spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals. |
| title | Recurrent subspaces in Banach spaces |
| topic | Functional Analysis Dynamical Systems 47A16, 47A53, 37B20 |
| url | https://arxiv.org/abs/2212.04464 |