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

Abstract

Let $\omega$ 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 $L^p(G,\omega )$ into itself. For a given set $\Gamma \subset \mathbb{C}$, a vector $f\in L^p(G,\omega )$ is said to be $(\Gamma ,S)$-dense if the set $\{\lambda T_{s}f:\lambda \in \Gamma ,s\in S\}$ is dense in $L^p(G,\omega )$. In this paper, we characterize the existence of $(\Gamma ,S)$-dense vectors in $L^p(G,\omega )$ in terms of the weight and the set $\Gamma$.

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