Essential Normality of automorphic composition operators

TL;DR

This paper characterizes when automorphic composition operators are essentially normal on various function spaces over the unit disk and ball, providing new insights into their spectral properties and examples of non-normal operators.

Contribution

It offers a comprehensive characterization of essentially normal automorphic composition operators on weighted Bergman and Hardy spaces, extending previous results and including new examples.

Findings

Characterization of essentially normal automorphic composition operators on $A^2_s(D)$

Analysis of automorphic composition operators on $H^2(B_N)$ and $A^2_s(B_N)$

Examples of composition operators induced by linear fractional maps that are not essentially normal

Abstract

We first characterize those composition operators that are essentially normal on the weighted Bergman space $A^2_s(D)$ for any real $s>-1$, where induced symbols are automorphisms of the unit disk $D$. Using the same technique, we investigate the automorphic composition operators on the Hardy space $H^2(B_N)$ and the weighted Bergman spaces $A^2_s(B_N)$ ($s>-1$). Furthermore, we give some composition operators induced by linear fractional self-maps of the unit ball $B_N$ that are not essentially normal.