Cesàro Means and Convex-Cyclic Operators

We characterize when the Cesàro means of higher order for Banach spaces operators are hypercyclic. This is a useful tool to prove that an operator is convex-cyclic and it provides a large number of examples of convex-cyclic operators. A complex number λ is said to be an extended eigenvalue of a bounded linear operator T if there exists a non-zero bounded linear operator X such that T X = λ X T . We will discover some necessary conditions on the extended spectrum of an operator to be a convex-cyclic operator. These conditions do not guarantee non-supercyclicity.