Invariant linear functionals on L^∞(R₊)

We consider a continuous version of the classical notion of Banach limits, i.e., normalized positive linear functionals on L^∞(R₊) invariant under translations f(x)↦f(x+s) of L^∞(R₊) for every s≥0. We give one of its characterizations in terms of the invariance under the operation of a certain linear transformation on L^∞(R₊). We also deal with invariant linear functionals under dilations f(x)↦f(rx), r≥1 and give a similar characterization via the Hardy operator. Applications to summability methods are presented in the last section.