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 \(T_{g,w}: f \mapsto w\cdot f*\delta_g\) 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.