Two families of hypercyclic nonconvolution operators (Q2037289)

A continuous linear operator \(T\) on a separable Fréchet space \(\mathcal{F}\) is hypercyclic if the orbit of some vector \(f \in \mathcal{F}\), defined by \(\{f, Tf, T^2f, \cdots \}\), is dense in \(\mathcal{F}\). Let \(H(\mathbb{C})\) denote the Fréchet space of all entire functions endowed with the topology of uniform convergence on compact sets. Some well-known examples of hypercyclic operators on \(H(\mathbb{C})\) are the translation operators \(T_a: f(z) \mapsto f(z +a)\) for \(a \neq 0\), and the derivative operator \(D: f(z) \mapsto f'(z)\). A convolution operator on \(H(\mathbb{C})\) is a continuous linear operator that commutes with all translations. It is known that nontrivial convolution operators are hypercyclic (see [\textit{G. Godefroy} and \textit{J. H. Shapiro}, J. Funct. Anal. 98, No. 2, 229--269 (1991; Zbl 0732.47016)]). For \(\lambda , b \in \mathbb{C}\), let \(C_{ \lambda , b} \) be the composition operator on \(H(\mathbb{C})\) defined by \(C_{ \lambda , b} f(z) = f(\lambda z + b)\). The study of hypercyclicity of non-convolution operators can be tracked back to the works of [\textit{R. Aron} and \textit{D. Markose}, J. Korean Math. Soc. 41, No. 1, 65--76 (2004; Zbl 1069.47006)] and [\textit{G. Fernández} and \textit{A. A. Hallack}, J. Math. Anal. Appl. 309, No. 1, 52--55 (2005; Zbl 1079.47010)] where they proved that a non-convolution operator \(C_{\lambda , b} \circ D\) is hypercyclic, if \(| \lambda | \geq 1\). The result is completed in [\textit{F. León-Saavedra} and \textit{P. Romero-de la Rosa}, Fixed Point Theory Appl. 2014, Paper No. 221, 5 p. (2014; Zbl 1334.47014)] by showing that this condition is necessary and sufficient for the hypercyclicity of \(C_{\lambda , b} \circ D\). In this paper, the authors have extended the previous result by discussing the hypercyclicity of two more general classes of operators. First, suppose that \(\psi\) is an entire function with \(\psi (0) = 0\), such that \(\psi (C_{\lambda , b} \circ D)\) is a continuous linear operator. It is shown that the necessary and sufficient condition for hypercyclicity of \(\psi (C_{\lambda , b} \circ D)\) is that \(| \lambda | \geq 1\). The sufficiency is proved by applying the hypercyclicity criterion, and the necessity is proved by showing that the orbit of any \(f \in H(\mathbb{C})\) under \(\psi (C_{\lambda , b} \circ D)\) tends to zero. Next, consider a non-constant entire function \(\psi\) of exponential type with \(\psi (0) = 0\) such that \(\psi (D)\) is a convolution operator. Applying the hypercyclicity criterion, it is shown that if \(| \lambda | \geq 1\) then \(C_{\lambda , b} \circ \psi (D)\) is hypercyclic.