# Baire theorem and hypercyclic algebras

Abstract : 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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Cited literature [33 references]

https://hal.archives-ouvertes.fr/hal-02308466
Contributor : Frédéric Bayart <>
Submitted on : Wednesday, September 16, 2020 - 1:55:17 PM
Last modification on : Thursday, September 17, 2020 - 3:31:19 AM

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### Identifiers

• HAL Id : hal-02308466, version 2
• ARXIV : 1910.05000

### Citation

Frédéric Bayart, Fernando Costa Júnior, Dimitris Papathanasiou. Baire theorem and hypercyclic algebras. 2020. ⟨hal-02308466v2⟩

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