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Abstract. Let X be a separable Banach space and M be a subspace of X. A bounded Linear operator T on X is subspace balanced convex-cyclic for a subspace M, if there exists a vector x∈X such that the intersection of balanced convex hull of orb(T,x) with M is dense in M. We give an example of subspace balanced convex-cyclic operator that is not balanced convex-cyclic. Also we give an improvement of the Kitailike criterion for subspace balanced convex-cyclicity and bring on with the Hahn-Banach characterization for subspace balanced convex-cyclicity.

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A bounded linear operator T on a locally convex space X is balanced convex-cyclic if there exists a vector x 2 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 f1; z; z²; z³; : : : g.In this paper we prove that the balanced convex-polynomials are dense in Lp() when ([-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.