Clark measures on the complex sphere

Abstract: Let $B_d$ denote the unit ball of $\mathbb{C}^d$, $d\ge 1$. Given a holomorphic function $\varphi: B_d \to B_1$, we study the corresponding family $σ_α[\varphi]$, $α\in\partial B_1$, of Clark measures on the unit sphere $\partial B_d$. If $\varphi$ is an inner function, then we introduce and investigate related unitary operators $U_α$ mapping analogs of model spaces onto $L^2(σ_α)$, $α\in\partial B_1$. In particular, we explicitly characterize the set of $U_α^* f$ such that $fσ_α$ is a pluriharmonic measure. Also, for an arbitrary holomorphic $\varphi: B_d \to B_1$, we use the family $σ_α[\varphi]$ to compute the essential norm of the composition operator $C_\varphi: H^2(B_1)\to H^2(B_d)$.