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.
Categories: math.FA