Trichotomy for the orbits of a hypercyclic operator on a Banach space
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910657032486912 |
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| author | Li, Jian |
| author_facet | Li, Jian |
| contents | We obtain a trichotomy for the orbits of a hypercyclic operator $T$ on a separable Banach space $X$: (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on $\ell^p$ show that all three cases can happen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trichotomy for the orbits of a hypercyclic operator on a Banach space Li, Jian Functional Analysis Dynamical Systems 47A16, 37B05 We obtain a trichotomy for the orbits of a hypercyclic operator $T$ on a separable Banach space $X$: (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on $\ell^p$ show that all three cases can happen. |
| title | Trichotomy for the orbits of a hypercyclic operator on a Banach space |
| topic | Functional Analysis Dynamical Systems 47A16, 37B05 |
| url | https://arxiv.org/abs/2407.07214 |