The local ergodic theorem and semigroups of nonpositive operators

Letting (T(t): t ⩾ 0) be a strongly continuous semigroup of contraction operators on L1(X,M,λ) we consider the question of the almost-everywhere convergence of (1α) ∝0α T(t)ƒ dt as α → 0 + (the local ergodic theorem). The limit is known to exist in the case where the semigroup is positive. In this paper we consider the question for general semigroups which may contain nonpositive elements. We show that a maximal ergodic inequality is equivalent to the local ergodic theorem. We also present some extensions to the local ergodic theorem. All the results of this paper also hold for N-parameter semigroups.