Norm preserving extensions of bounded holomorphic functions

Abstract

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 $\varphi$ on $\Omega$ that agrees with $p$ on $V$ and whose $H^{\infty }$ norm on $\Omega$ equals the sup-norm of $p$ on $V$. We show that if $\Omega$ is either strictly convex or strongly linearly convex in ${\mathbb{C}}^2$, 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.