Density properties of orbits for a hypercyclic operator on a Banach space
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908792791236608 |
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| author | Li, Jian Wang, Xinsheng Zhao, Jianjie |
| author_facet | Li, Jian Wang, Xinsheng Zhao, Jianjie |
| contents | We study density properties of orbits for a hypercyclic operator $T$ on a separable Banach space $X$, and show that exactly one of the following four cases holds: (1) every vector in $X$ is asymptotic to zero with density one; (2) generic vectors in $X$ are distributionally irregular of type $1$; (3) generic vectors in $X$ are distributionally irregular of type $2\frac{1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$; (4) every hypercyclic vector in $X$ is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$-semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density properties of orbits for a hypercyclic operator on a Banach space Li, Jian Wang, Xinsheng Zhao, Jianjie Functional Analysis We study density properties of orbits for a hypercyclic operator $T$ on a separable Banach space $X$, and show that exactly one of the following four cases holds: (1) every vector in $X$ is asymptotic to zero with density one; (2) generic vectors in $X$ are distributionally irregular of type $1$; (3) generic vectors in $X$ are distributionally irregular of type $2\frac{1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$; (4) every hypercyclic vector in $X$ is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$-semigroups. |
| title | Density properties of orbits for a hypercyclic operator on a Banach space |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.19752 |