On convex-cyclic operators

We give a Hahn-Banach characterization for convex-cyclicity. We also obtain an example of a bounded linear operator S on a Banach space with σp(S⁎)=∅ such that S is convex-cyclic, but S is not weakly hypercyclic and S2 is not convex-cyclic. This solved two questions of Rezaei in [25] when σp(S⁎)=∅. We also characterize the diagonalizable normal operators that are convex-cyclic and give a condition on the eigenvalues of an arbitrary operator for it to be convex-cyclic. We show that certain adjoint multiplication operators are convex-cyclic and show that some are convex-cyclic but no convex polynomial of the operator is hypercyclic. Also some adjoint multiplication operators are convex-cyclic but not 1-weakly hypercyclic.