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Article Dans Une Revue Advances in Mathematics Année : 2021

Baire theorem and hypercyclic algebras

Résumé

The question of whether a hypercyclic operator $T$ acting on a Fréchet algebra $X$ admits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterizations in the context of convolution operators acting on $H(\mathbb C)$ and backward shifts acting on a general Fréchet sequence algebra. Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighted backward shift acting on $c_0$ admitting a frequently hypercyclic algebra for the coordinatewise product. The closed hypercyclic algebra problem is also covered.
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Dates et versions

hal-02308466 , version 1 (09-10-2019)
hal-02308466 , version 2 (16-09-2020)

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Frédéric Bayart, Fernando Costa Júnior, Dimitris Papathanasiou. Baire theorem and hypercyclic algebras. Advances in Mathematics, 2021, 376, pp.107419. ⟨10.1016/j.aim.2020.107419⟩. ⟨hal-02308466v2⟩
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