Commutators of automorphic composition operators with adjoints

Abstract

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

15 pages