Powers of convex-cyclic operators (Q1724612)

Summary: A bounded operator \(T\) on a Banach space \(X\) is convex cyclic if there exists a vector \(x\) such that the convex hull generated by the orbit \(\left\{T^n x\right\}_{n \geq 0}\) is dense in \(X\). In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operator \(T\) such that the power \(T^n\) fails to be convex cyclic. Using this result we solve three questions posed by \textit{H. Rezaei} [Linear Algebra Appl. 438, No. 11, 4190--4203 (2013; Zbl 1311.47012)].