Convex-Cyclic Weighted Translations On Locally Compact Groups

TL;DR

This paper investigates conditions under which weighted translation operators on locally compact groups are convex-cyclic, providing sufficient and necessary conditions, supported by examples.

Contribution

It introduces new criteria for convex-cyclicity of weighted translation operators on locally compact groups, expanding understanding of operator dynamics in this setting.

Findings

Sufficient conditions for convex-cyclicity of weighted translation operators.

Necessary conditions for convex-cyclicity.

Examples illustrating the theoretical results.

Abstract

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.