Composition of inner mappings on the ball

Abstract:

Suppose that $F:B_k\to B_m$ is an inner map and that $G\in H^\infty (B_m)^n$. We show that the identity \[ (G\circ F)^\ast =r(G)\circ F^\ast \] holds with an abstract boundary value $r(G)$. If the natural compatibility condition $\sigma _k^{F^\ast }\ll \sigma _m$ is satisfied, then $r(G)=G^\ast$. Here, $\sigma _k^{F^\ast }$ denotes the image of the surface measure on $S_k$ under $F^\ast$. In particular, $G\circ F$ is inner if $F$ and $G$ are inner and $\sigma _k^{F^\ast }\ll \sigma _m$. Furthermore, we characterize the boundedness of composition operators on Hardy spaces in terms of the absolute continuity of $\sigma _k^{F^\ast }$.