On composition operators on Hilbert spaces of entire functions

We obtain a complete description of the families of continuous and compact composition operators on Hilbert spaces of entire functions introduced by K. Chan and J. Shapiro to study some dynamical properties of translation operators. In contrast to recent papers devoted to the same problems we do not use any additional assumptions on the spaces. We apply a new approach based on the embedding of a Hilbert space under consideration into some appropriate Banach space with sup-norm.