A group structure on D and its application for composition operators

Abstract

We present a group structure on $\mathbb{D}$ via the automorphisms which fix the point $1$. Through the induced group action, each point of $\mathbb{D}$ produces an equivalence class that turns out to be a Blaschke sequence. We show that the corresponding Blaschke products are minimal/atomic solutions of the functional equation $\psi \circ \phi =\lambda \psi$, where $\lambda$ is a unimodular constant and $\phi$ is an automorphism of the unit disk. We also characterize all Blaschke products that satisfy this equation, and we study its application in the theory of composition operators on model spaces $K_{\Theta }$.