Powers of Convex-Cyclic Operators

Abstract

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 ${\left\{T^{n}x\right\}}_{n\ge 0}$ 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 Rezaei (2013).