Ergodic properties of contraction semigroups in L p , 1 < p < ∞

Let {T (t) : t > 0} be a strongly continuous semigroup of linear contractions in Lp, 1 < p < ∞, of a σ-finite measure space. In this paper we prove that if there corresponds to each t > 0 a positive linear contraction P (t) in Lp such that |T (t)f | ≤ P (t)|f | for all f ∈ Lp, then there exists a strongly continuous semigroup {S(t) : t > 0} of positive linear contractions in Lp such that |T (t)f | ≤ S(t)|f | for all t > 0 and f ∈ Lp. Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in Lp, we also prove multiparameter pointwise ergodic and local ergodic theorems for such semigroups.