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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 $σ_{p}(S^*)=\emptyset$ such that $S$ is convex-cyclic, but $S$ is not weakly hypercyclic and $S^2 $ is not convex-cyclic. This solved two questions of Rezaei in \cite{Rezaei} when $σ_p(S^*)=\varnothing$. %Recently, León-Saavedra and Romero de la Rosa \cite{LeRo} provide an example of a convex-cyclic operator $S$ such that the power $S^n$ fails to be convex-cyclic with $σ_p(S^*)\neq \varnothing$. In fact they solved tree questions posed by Rezaei in \cite{Rezaei}. Moreover, we prove that $m$-isometries are not convex-cyclic and that $\varepsilon$-hypercyclic operators are convex-cyclic. 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.