On the convex hull generated by orbit of operators

Let X be an infinite dimensional Banach space and T be a bounded operator on X. The operator T is said to be convex-cyclic if the convex hull generated by orb(T,x) is dense in X for some x∈X. We characterize completely such operators on finite dimensional vector spaces. Some examples of pure convex-cyclic operators on infinite dimensional Banach spaces are provided and many hypercyclic-like properties of such operators are considered.