Group invariant separating polynomials on a Banach space

Abstract

We study the group invariant continuous polynomials on a Banach space $X$ that separate a given set $K$ in $X$ and a point $z$ outside $K$. We show that if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L} (X)$, $K$ is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be separated from $K$ by a continuous polynomial $Q$, then $z$ can also be separated from $K$ by a $G$-invariant continuous polynomial $P$. It turns out that this result does not hold when $X$ is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that $X$ has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.