Convex-Cyclic Weighted Translations On Locally Compact Groups

A bounded linear operator $T$ on a Banach space $X$ is called a convex-cyclic operator if there exists a vector $x\in X$ such that the convex hull of $Orb(T,x)$ is dense in $X$. In this paper, for given an aperiodic element $g$ in a locally compact group $G$, we give some sufficient conditions for a weighted translation operator $T_{g,w}:f\mapsto w\cdot f\ast {\delta }_g$ on ${\mathfrak{L}}^p(G)$ to be convex-cyclic. A necessary condition is also studied. At the end, to explain the obtained results, some examples are given.