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Commutators of automorphic composition operators with adjoints

In this paper, we investigate the compactness of the commutator $[C_ψ^{\ast}, C_φ]$ on the Hardy space $H^2(B_N)$ or the weighted Bergman space $A^2_s(B_N)$ ($s>-1$), when $φ$ and $ψ$ are automorphisms of the unit ball $B_N$. We obtain that $[C_ψ^{\ast}, C_φ]$ is compact if and only if $φ$ and $ψ$ 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 $[C_ψ^{\ast}, C_φ]$ on the Dirichlet space $\mathcal{D}(B_N)$, where $φ$ and $ψ$ are linear fractional self-maps or both automorphisms of $B_N$.