Convex-cyclic weighted translations on locally compact groups

Abstract

A bounded linear operator T on a Banach space X is called a convex-cyclic operator if there exists a vector x ∈ X such that the convex hull of O r b ( 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 ↦ w ⋅ f ∗ δ g on 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.