Invariant linear functionals on $L^{\infty }({\mathbb{R}}_{+})$

Abstract

We consider a continuous version of the classical notion of Banach limits, namely, positive linear functionals on $L^{\infty }({\mathbb{R}}_{+})$ invariant under translations $f(x)\mapsto f(x+s)$ of $L^{\infty }({\mathbb{R}}_{+})$ for every $s\ge 0$. We give its characterization in terms of the invariance under the operation of a certain linear mapping on $L^{\infty }({\mathbb{R}}_{+})$. Applications to summability methods are provided in the last section.

19 pages