On frequently supercyclic operators and an F_Γ-hypercyclicity criterior with applications
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| Format: | Preprint |
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2024
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| _version_ | 1866917829130846208 |
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| author | Alves, Thiago R. Botelho, Geraldo Fávaro, Vinicius V. |
| author_facet | Alves, Thiago R. Botelho, Geraldo Fávaro, Vinicius V. |
| contents | Given a Furstenberg family F and a subset Γ of C, we introduce and explore the notions of F_Γ-hypercyclic operator and F-hypercyclic scalar set. First, the study of F_C-hypercyclic operators yields new interesting information about frequently supercyclic, U-frequently supercyclic, reiteratively supercyclic and supercyclic operators. Then we provide a criterion for identifying F_Γ-hypercyclic operators. As applications of this criterion, we show that any unilateral pseudo-shift operator on c_0(N) or l_p(N) is F_Γ-hypercyclic for every unbounded subset Γ of C. Moreover, under the same condition on Γ, we show that any separable infinite-dimensional Banach space supports an F_Γ-hypercyclic operator. Finally, our study provides sufficient and necessary conditions for a subset Γ of C to be a hypercyclic scalar set. These results give partial answers to a question raised by Charpentier, Ernst, and Menet in 2016. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On frequently supercyclic operators and an F_Γ-hypercyclicity criterior with applications Alves, Thiago R. Botelho, Geraldo Fávaro, Vinicius V. Functional Analysis 47A16, 47B01 Given a Furstenberg family F and a subset Γ of C, we introduce and explore the notions of F_Γ-hypercyclic operator and F-hypercyclic scalar set. First, the study of F_C-hypercyclic operators yields new interesting information about frequently supercyclic, U-frequently supercyclic, reiteratively supercyclic and supercyclic operators. Then we provide a criterion for identifying F_Γ-hypercyclic operators. As applications of this criterion, we show that any unilateral pseudo-shift operator on c_0(N) or l_p(N) is F_Γ-hypercyclic for every unbounded subset Γ of C. Moreover, under the same condition on Γ, we show that any separable infinite-dimensional Banach space supports an F_Γ-hypercyclic operator. Finally, our study provides sufficient and necessary conditions for a subset Γ of C to be a hypercyclic scalar set. These results give partial answers to a question raised by Charpentier, Ernst, and Menet in 2016. |
| title | On frequently supercyclic operators and an F_Γ-hypercyclicity criterior with applications |
| topic | Functional Analysis 47A16, 47B01 |
| url | https://arxiv.org/abs/2411.03179 |