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 $\phi$ is an involutive Moebius automorphism of ${\mathbb{B}}_n$, we find a conjugation operator $\mathcal{J}$ on $\mathcal{H}$ such that $C_{\phi }=\mathcal{J}{C}_{\phi }^{\ast }\mathcal{J}$. The case $n=1$ answers a question of Garcia and Hammond.

5 pages

References in corpus (2)

• Complex symmetric partial isometries
• Self-adjoint, unitary, and normal weighted composition operators in several variables