$Γ$-supercyclicity of families of translates in weighted $L^p$-spaces on locally compact groups

Abstract

Let $ω$ be a weight function defined on a locally compact group $G$, $1\le p<+\infty$, $S\subset G$ and let us assume that for any $s\in S$, the left translation operator $T_s$ is continuous from the weighted $L^p$-space $L^p(G,ω)$ into itself. For a given set $Γ\subset\mathbb{C}$, a vector $f\in L^p(G,ω)$ is said to be $(Γ,S)$-dense if the set $\{ λT_sf:\, λ\in Γ, \,s\in S\}$ is dense in $L^p(G,ω)$. In this paper, we characterize the existence of $(Γ,S)$-dense vectors in $L^p(G,ω)$ in terms of the weight and the set $Γ$.