Eigenfunctions of Composition Operators on Bloch-type Spaces

TL;DR

This paper investigates conditions under which solutions to the Schr"oder equation for certain composition operators on the unit disk belong to Bloch-type spaces, extending results to weighted composition operators.

Contribution

It provides new sufficient conditions for solutions to be in Bloch-type spaces and extends these results from composition to weighted composition operators.

Findings

Conditions ensuring solutions are in Bloch-type spaces

Extension of results to weighted composition operators

Characterization of solutions for specific holomorphic self-maps

Abstract

Suppose $\varphi$ is a holomorphic self map of the unit disk and $C_\varphi$ is a composition operator with symbol $\varphi$ that fixes the origin and $0<|\varphi'(0)|<1$. This work explores sufficient conditions that ensure all holomorphic solutions of Schr\"oder equation for the composition operator $C_\varphi$ belong to a Bloch-type space $\mathcal{B}_\alpha$ for some $\alpha>0$. The results from composition operators have been extended to weighted composition operators in the second part of this work.