Typicality of operators on Fréchet algebras admitting a hypercyclic algebra
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913901913833472 |
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| author | Alexandre, William Gilmore, Clifford Grivaux, Sophie |
| author_facet | Alexandre, William Gilmore, Clifford Grivaux, Sophie |
| contents | This paper is devoted to the study of typical properties (in the Baire Category sense) of certain classes of continuous linear operators acting on Fréchet algebras, endowed with the topology of pointwise convergence. Our main results show that within natural Polish spaces of continuous operators acting on the algebra $H(\mathbb{C})$ of entire functions on $\mathbb{C}$, a typical operator supports a hypercyclic algebra. We also investigate the case of the complex Fréchet algebras $X=\ell_{p}(\mathbb{N})$, $1\le p<+\infty$, or $X=c_{0}(\mathbb{N})$ endowed with the coordinatewise product, and show that whenever $M>1$, a typical operator on $X$ of norm less than or equal to $M$ admits a hypercyclic algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_08321 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Typicality of operators on Fréchet algebras admitting a hypercyclic algebra Alexandre, William Gilmore, Clifford Grivaux, Sophie Functional Analysis 7A16, 46B87, 46E05, 54E52 This paper is devoted to the study of typical properties (in the Baire Category sense) of certain classes of continuous linear operators acting on Fréchet algebras, endowed with the topology of pointwise convergence. Our main results show that within natural Polish spaces of continuous operators acting on the algebra $H(\mathbb{C})$ of entire functions on $\mathbb{C}$, a typical operator supports a hypercyclic algebra. We also investigate the case of the complex Fréchet algebras $X=\ell_{p}(\mathbb{N})$, $1\le p<+\infty$, or $X=c_{0}(\mathbb{N})$ endowed with the coordinatewise product, and show that whenever $M>1$, a typical operator on $X$ of norm less than or equal to $M$ admits a hypercyclic algebra. |
| title | Typicality of operators on Fréchet algebras admitting a hypercyclic algebra |
| topic | Functional Analysis 7A16, 46B87, 46E05, 54E52 |
| url | https://arxiv.org/abs/2312.08321 |