Convolution operators supporting hypercyclic algebras (Q313494)

Let \(\Omega\subset\mathbb{C}\) be a simply connected domain and let \(H(\Omega)\) be the space of functions holomorphc in \(\Omega\), endowed with the topology of uniform convergence on compacta. The authors consider the operator \(\Phi(D)\), where \(\Phi\) is a polynomial that vanishes at zero and \(D:H(\Omega)\to H(\Omega)\), \(f\mapsto f'\), is the differentiation operator. For a suitably defined set \(\mathcal{A}\subset H(\Omega)\), they prove thatNEWLINENEWLINENEWLINE{\parindent=0.7cmNEWLINE\begin{itemize}\item[(i)] the set \(\mathcal{A}\) is a \(G_{\delta}\) and dense subset of \(H(\Omega)\);NEWLINE\item[(ii)] every function \(f\in\mathcal{A}\) generates an algebra whose non-zero elements are all hypercyclic for the operator \(\Phi(D)\).NEWLINENEWLINE\end{itemize}}NEWLINEThis work generalizes the work of \textit{F. Bayart} and \textit{É. Matheron} (Theorem 8.26 in [Dynamics of linear operators. Cambridge: Cambridge University Press (2009; Zbl 1187.47001)]).