Hypercyclic convolution operators on spaces of entire functions (Q2835253)

The authors show that no convolution operator on the space \(H(\mathbb{C}^{\mathbb{N}})\) of entire functions on the countable product \(\mathbb{C}^{\mathbb{N}}\) of copies of the complex plane is hypercyclic. This result is in contrast with the fact proved by \textit{G. Godefroy} and \textit{J. H. Shapiro} [J. Funct. Anal. 98, No.~2, 229--269 (1991; Zbl 0732.47016)] stating that every nontrivial convolution operator on the space of entire functions of several complex variables is hypercyclic. On the other hand, the authors prove that, if \(F\) is a separable Fréchet space with the approximation property, in particular, if \(F\) is nuclear Fréchet space, and \(E\) denotes the dual of \(F\) endowed with the topology of uniform convergence on the compact subsets of \(F\), then every nontrivial convolution operator on the space \(H(E)\) of entire functions on \(E\) is hypercyclic.