CONVEX STONE-WEIERSTRASS THEOREMS AND INVARIANT CONVEX SETS

A convex polynomial is a convex combination of the monomials {1, x, x(2), ... }. This paper establishes that the convex polynomials on R are dense in L-p(mu) and weak* dense in L-infinity(mu) whenever mu is a compactly supported regular Borel measure on R and mu([-1, infinity)) = 0. It is also shown that the convex polynomials are norm dense in C(K) precisely when K boolean AND [-1, infinity) = empty set, 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 convex-cyclic if there is a vector x is an element of 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.