Ultra hypercyclicity and its connection to mixing properties
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911275541331968 |
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| author | Liu, Martin Walmsley, David Xue, James |
| author_facet | Liu, Martin Walmsley, David Xue, James |
| contents | Recently, two new topological properties for operators acting on a topological vector space were introduced: strong hypercyclicity and hypermixing. We introduce a new property called ultra hypercyclicity and compare it to strong hypercyclicity and hypermixing, as well as the classical notions of mixing, weak mixing, and hypercyclicity. We show that every ultra hypercyclic operator on Fréchet space must be weakly mixing, and that there exists a strongly hypercyclic operator which is not ultra hypercyclic. We also characterize, in terms of the weight sequence, the ultra hypercyclic weighted backward shifts on $c_0$ and $\ell^p$, $1\leq p<\infty$. Finally, we improve upon a necessary condition for strongly hypercyclic weighted backward shifts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02899 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ultra hypercyclicity and its connection to mixing properties Liu, Martin Walmsley, David Xue, James Functional Analysis Recently, two new topological properties for operators acting on a topological vector space were introduced: strong hypercyclicity and hypermixing. We introduce a new property called ultra hypercyclicity and compare it to strong hypercyclicity and hypermixing, as well as the classical notions of mixing, weak mixing, and hypercyclicity. We show that every ultra hypercyclic operator on Fréchet space must be weakly mixing, and that there exists a strongly hypercyclic operator which is not ultra hypercyclic. We also characterize, in terms of the weight sequence, the ultra hypercyclic weighted backward shifts on $c_0$ and $\ell^p$, $1\leq p<\infty$. Finally, we improve upon a necessary condition for strongly hypercyclic weighted backward shifts. |
| title | Ultra hypercyclicity and its connection to mixing properties |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2312.02899 |