Common hypercyclic algebras for families of products of backward shifts

TL;DR

This paper extends the theory of common hypercyclic vectors to algebras, demonstrating the existence of common hypercyclic algebras for families of backward shift operators, including a positive answer to a multidimensional question.

Contribution

It generalizes recent results to algebraic contexts and provides a multidimensional example of a common hypercyclic algebra for weighted backward shifts.

Findings

Existence of common hypercyclic algebras for families of backward shift operators.

Positive answer to a multidimensional hypercyclic algebra question.

Construction of a common hypercyclic algebra on ll_1(bN) with convolution product.

Abstract

In this paper, we generalize to the context of algebras some recent results on the existence of common hypercyclic vectors for families of products of backward shift operators. We also give, in a multi-dimensional setting, a positive answer to a question raised by F. Bayart, D. Papathanasiou and the author about the existence of a common hypercyclic algebra on $\ell_1(\mathbb{N})$ with the convolution product for the family of backward shifts $(B_{w(\lambda)})_{\lambda>0}$ induced by the weights $w_n(\lambda)=1+\lambda/n$.