Convex Stone-Weierstrass theorems and invariant convex sets

Abstract

A \textit {convex polynomial} is a convex combination of the monomials $\{1, x, x^2, \ldots \}$. This paper establishes that the convex polynomials on $\mathbb R$ are dense in $L^p(\mu )$ and weak$^*$ dense in $L^\infty (\mu )$ whenever $\mu $ is a compactly supported regular Borel measure on $\mathbb {R}$ and $\mu ([-1,\infty )) = 0$. It is also shown that the convex polynomials are norm dense in $C(K)$ precisely when $K \cap [-1, \infty ) = \emptyset $, where $K$ is a compact subset of the real line. Moreover, the closure of the convex polynomials on $[-1,b]$ is shown to be the functions that have a convex power series representation.

A continuous linear operator $T$ on a locally convex space $X$ is \textit {convex-cyclic} if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.