2015/10/29 by Nathan S. Feldman, Feldman, Nathan S., Paul J. McGuire +1
Mathematics · #47A16 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:47A16
paper · pdf · doi:10.48550/arxiv.1510.08878
arxiv created 2015/10/29 · arxiv updated 2015/11/02
A convex-polynomial is a convex combination of the monomials \1, x, x2, …\. This paper establishes that the convex-polynomials on \mathbb R are dense in Lp(μ) and weak^* dense in L^∞(μ), precisely when μ([-1,∞)) = 0. It is shown that the convex-polynomials are dense in C(K) precisely when K ∩ [-1, ∞) = ∅, where K is a compact subset of the real line. Moreover, the closure of the convex-polynomials on [-1,b] are 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 ∈ X such that the convex hull of the orbit of x is dense in X. The above results characterize which multiplication operators on various real Banach spaces are convex-cyclic. It is shown for certain multiplication operators that every closed invariant convex set is a closed invariant subspace.