On the convergence of iterates of convolution operators in Banach spaces

Let G be a locally compact abelian group and let M(G) be the measure algebra of G. A measure μ∈M(G) is said to be power bounded if supn≥0∥μn∥1<∞. Let T={Tg:g∈G} be a bounded and continuous representation of G on a Banach space X. For any μ∈M(G), there is a bounded linear operator on X associated with µ, denoted by Tμ, which integrates Tg with respect to µ. In this paper, we study norm and almost everywhere behavior of the sequences {Tnμx} (x∈X) in the case when µ is power bounded. Some related problems are also discussed.