Commutators of automorphic composition operators with adjoints

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 $B_N$.