Complex symmetry of Composition operators induced by involutive Ball automorphisms

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 $C_ψ$ defined by $C_ψf=f\circψ$ is bounded on $\mathcal{H}$ whenever $ψ$ is a linear fractional self-map of $\mathbb{B}_n$. If $φ$ is an involutive Moebius automorphism of $\mathbb{B}_n$, we find a conjugation operator $\mathcal{J}$ on $\mathcal{H}$ such that $C_φ=\mathcal{J} C^*_φ\mathcal{J}$. The case $n=1$ answers a question of Garcia and Hammond.

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