Complex symmetry of composition operators induced by involutive ball automorphisms

Abstract:

Suppose $\mathcal {H}$ is a weighted Hardy space of analytic functions on the unit ball $\mathbb {B}_n\subset \mathbb {C}^n$ such that the composition operator $C_\psi$ defined by $C_{\psi }f=f\circ \psi$ is bounded on $\mathcal {H}$ whenever $\psi$ is a linear fractional self-map of $\mathbb {B}_n$. If $\varphi$ is an involutive Moebius automorphism of $\mathbb {B}_n$, we find a conjugation operator $\mathcal {J}$ on $\mathcal {H}$ such that $C_{\varphi }=\mathcal {J} C^*_{\varphi }\mathcal {J}$. The case $n=1$ answers a question of Garcia and Hammond.