NORM PRESERVING EXTENSIONS OF BOUNDED HOLOMORPHIC FUNCTIONS

A relatively polynomially convex subset V of a domain Omega has the extension property if for every polynomial p there is a bounded holomorphic function phi on Omega that agrees with p on V and whose H-infinity norm on Omega equals the sup-norm of p on V. We show that if Q is either strictly convex or strongly linearly convex in C-2, or the ball in any dimension, then the only sets that have the extension property are retracts. If Omega is strongly linearly convex in any dimension and V has the extension property, we show that V is a totally geodesic submanifold. We show how the extension property is related to spectral sets.