Topological and Dynamical Properties of Composition Operators

Abstract

We study various properties of composition operators $C_\psi$ acting between generalized Fock spaces ${\mathcal {F}}_\varphi ^p$ and ${\mathcal {F}}_\varphi ^q$ with weight functions $\varphi$ growing faster than the classical Gaussian function $\frac{1}{2}|z|^2$ and satisfy some mild smoothness conditions. We show that if $p\ne q,$ then $C_\psi :{\mathcal {F}}_\varphi ^p \rightarrow {\mathcal {F}}_\varphi ^q$ is bounded if and only if it is compact. This result shows a significance difference from the analogous result for the case when $C_\psi$ acts between the classical Fock spaces or generalized Fock spaces where the weight functions grow slower than the Gaussian function. We further described the Schatten ${\mathcal {S}}_p({\mathcal {F}}_\varphi ^2)$ class, hyponormal, unitary, cyclic and supercyclic composition operators on the spaces. As an application, we also characterized the compact differences, the isolated and essentially isolated points, and connected components of the space of the operators under the operator norm topology.