Algebrability of the set of hypercyclic vectors for backward shift operators (Q2309112)

Let \(X\) be a Fréchet space and \(T: X\to X\) be a continuous linear map. The set of all hypercyclic vectors of \(T\) is denoted by \[ HC(T)=\{x\in X: \{x,Tx,T^2x,\dots\}\text{ is dense in }X\}. \] By the well-known Herrero-Bourdon theorem, the set \(HC(T)\) is either empty or contains a dense linear subspace (but the origin). When \(T\) is a hypercyclic operator on a Fréchet algebra \(X\) it is then natural to ask whether \(HC(T)\) contains a non-trivial subalgebra of \(X\) (except zero). When such a subalgebra exists it is called a hypercyclic algebra for \(T\). If \(T\) admits a hypercyclic algebra which is not finitely generated then \(HC(T)\) is said to be algebrable. In this direction, \textit{S. Shkarin} [Isr. J. Math. 180, 271--283 (2010; Zbl 1218.47017)] and \textit{F. Bayart} et al. [Dynamics of linear operators. Cambridge: Cambridge University Press (2009; Zbl 1187.47001)] showed independently that the differentiation operator \(D\) acting on \(H(\mathbb{C})\) admits a hypercyclic algebra, thereby providing the first known example of an operator that admits a hypercyclic algebra. Using the ideas of \textit{J. Bès} et al. [J. Math. Anal. Appl. 445, No. 2, 1232--1238 (2017; Zbl 1384.47002)] extended such a result to convolution operators \(P(D)\) induced by non-constant polynomials \(P\) that vanish at zero. In the paper under review, the authors improve the result of Shkarin, Bayart and Matheron on the differentiation operator \(D\) in two ways. First, they consider general weighted backward shift operators on Fréchet sequence algebras, where the multiplicative structure can be given either by coordinatewise multiplication or by the Cauchy product (that is, the discrete convolution). And secondly, they obtain in each case that the set of hypercyclic vectors is algebrable.