Complex symmetry of composition operators induced by involutive ball automorphisms

Abstract

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