# Topological and dynamical properties of composition operators

## TL;DR

This paper investigates the properties of composition operators on generalized Fock spaces with rapidly growing weights, revealing new boundedness and compactness criteria and analyzing their spectral and topological characteristics.

## Contribution

It establishes that for different p and q, boundedness implies compactness of composition operators on these spaces, and characterizes various classes and topological features of these operators.

## Key findings

- Boundedness and compactness are equivalent for composition operators when p ≠ q.
- Characterization of Schatten class, normal, unitary, cyclic, and supercyclic composition operators.
- Topological analysis of the space of operators, including isolated points and connected components.

## Abstract

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