Composition of inner mappings on the ball

Suppose that F : B-k --> B-m is an inner map and that G is an element of H- infinity (B-m)(n). We show that the identity (G circle F)* = r(G) circle F* holds with an abstract boundary value r(G). If the natural compatibility co ndition sigma (F*)(k) much less than sigmam is satisfied, then r(G) =G*. He re, sigma (F*)(k) denotes the image of the surface measure on S-k under F*. Inparticular,G circle F is inner if F and G are inner and sigma (F*)(k) mu ch less than sigma (m). Furthermore, we characterize the boundedness of com position operators on Hardy spaces in terms of the absolute continuity of s igma (F*)(k).