At the crossroads of holomorphic dynamic and operator theory: spectral properties of composition operators on $\mathrm {Hol}(\mathbb {B}_N)$

Abstract We study the class of composition operators acting on the Fréchet space $\mathrm {Hol}(\mathbb {B}_N)$ of all holomorphic maps on the unit ball of $\mathbb {C}^N$ . We describe the conditions to make these operators continuous, invertible and compact. We also do the spectral study of these operators, depending on the nature of its symbol.