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arXiv · 2411.14609

Hypercyclic algebras for weighted shifts on trees

Arafat Abbar·Fernando Costa Jr
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Abstract

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When $V$ is a rooted directed tree, we show the set of hypercyclic vectors of any backward weighted shift operator on the space $c_0(V)$ or $\ell^1(V)$ is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on $\ell^p(V), 1<p<+\infty$. Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing backward weighted shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.

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Arafat Abbar, Fernando Costa Jr. 2024-11-21. Hypercyclic algebras for weighted shifts on trees. https://arxiv.org/abs/2411.14609

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