On frequently supercyclic operators and an F_{\Gamma}-hypercyclicity criterior with applications

Thiago R. Alves; Geraldo Botelho; Vinicius V. Fávaro

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arXiv · 2411.03179

On frequently supercyclic operators and an F_{\Gamma}-hypercyclicity criterior with applications

Thiago R. Alves·Geraldo Botelho·Vinicius V. Fávaro

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Abstract

Given a Furstenberg family F and a subset {\Gamma} of C, we introduce and explore the notions of F_{\Gamma}-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_{\Gamma}-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_{\Gamma}-hypercyclic for every unbounded subset {\Gamma} of C. Moreover, under the same condition on {\Gamma}, we show that any separable infinite-dimensional Banach space supports an F_{\Gamma}-hypercyclic operator. Finally, our study provides sufficient and necessary conditions for a subset {\Gamma} of C to be a hypercyclic scalar set. These results give partial answers to a question raised by Charpentier, Ernst, and Menet in 2016.

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Thiago R. Alves, Geraldo Botelho, Vinicius V. Fávaro. 2024-11-05. On frequently supercyclic operators and an F_{\Gamma}-hypercyclicity criterior with applications. https://arxiv.org/abs/2411.03179

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