Norm preserving extensions of bounded holomorphic functions

TL;DR

This paper characterizes subsets with the extension property in certain convex domains in complex spaces, showing they are either retracts or totally geodesic submanifolds, and explores their relation to spectral sets.

Contribution

It establishes that in specific convex domains, the only sets with the extension property are retracts or totally geodesic submanifolds, providing a geometric characterization.

Findings

Sets with the extension property are retracts in strictly convex or strongly linearly convex domains in ${f C}^2$ and the ball.

In strongly linearly convex domains, extension property implies the set is a totally geodesic submanifold.

The extension property is connected to the concept of spectral sets.

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 $\phi$ 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.