On a theorem of Godefroy and Shapiro
The authors generalize a result of \textit{G.\,Godefroy} and \textit{J.\,H.\,Shapiro} [J.~Funct.\ Anal.\ 98, No.\,2, 229--269 (1991; Zbl 0732.47016)], namely, let \(T\) be a continuous linear operator on the space \(H(\mathbb{C}^N)\) of holomorphic functions on \(\mathbb{C}^N\), \(N\geq 1\), which commutes with all translation operators \(T_a: f\mapsto f(.+a)\), \(a \in \mathbb{C}^N\), and is not a scalar multiple of the identity operator. Then \(T\) is frequently hypercyclic, i.e., there exists a function \(f \in H(\mathbb{C}^N)\) such that for every nonempty open subset \(U\) of \(H(\mathbb{C}^N)\), the set \(\{n\in\mathbb{N} \mid T^n x \in U\}\) has positive lower density (there even exists a \(T\)-invariant ergodic Gaussian measure on \(H(\mathbb{C}^N)\) whose support is the whole space). The rate of growth of such frequently hypercyclic functions at infinity is also studied.
scientific article; zbMATH DE number 5071134