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