Commutators of automorphic composition operators with adjoints

TL;DR

This paper characterizes when the commutator of automorphic composition operators and their adjoints is compact on various function spaces, showing it occurs precisely when the automorphisms commute and are unitary, extending known results to multiple variables.

Contribution

It provides a complete characterization of the compactness of the commutator of automorphic composition operators on several spaces, generalizing previous one-variable results with a new, simpler approach.

Findings

The commutator is compact iff automorphisms commute and are unitary.

Extension of results from one-variable to several variables.

Simplified technique for analyzing operator compactness.

Abstract

In this paper, we investigate the compactness of the commutator $[C_\psi^{\ast}, C_\varphi]$ on the Hardy space $H^2(B_N)$ or the weighted Bergman space $A^2_s(B_N)$ ($s>-1$), when $\varphi$ and $\psi$ are automorphisms of the unit ball $B_N$. We obtain that $[C_\psi^{\ast}, C_\varphi]$ is compact if and only if $\varphi$ and $\psi$ commute and they are both unitary. This generalizes the corresponding result in one variable. Moreover, our technique is different and simpler. In addition, we also discuss the commutator $[C_\psi^{\ast}, C_\varphi]$ on the Dirichlet space $\mathcal{D}(B_N)$, where $\varphi$ and $\psi$ are linear fractional self-maps or both automorphisms of $B_N$.