Density of Balanced Convex-polynomials In LP(μ)

A bounded linear operator T on a locally convex space X is balanced convexcyclic if there exists a vector x ∈ X such that the balanced convex hull of orb(T, x) is dense in X.A balanced convex-polynomial is a balanced convex combination of monomials {1, z, z2 , z3 , . . . }.In this paper we prove that the balanced convex-polynomials are dense in L p (µ) when µ([−1, 1]) = 0. Our results are used to characterize which multiplication operators on various real Banach spaces are balanced convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant balanced convex-set is a closed invariant subspace