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

TL;DR

This paper characterizes positive linear functionals on $L^{ obreakdash}^{ ext{infty}}( obreakdash ext{R}_+)$ that are invariant under translations, extending the classical Banach limits concept with applications to summability methods.

Contribution

It provides a new characterization of invariant linear functionals on $L^{ ext{infty}}( ext{R}_+)$ using invariance under a specific linear map, extending Banach limits.

Findings

Characterization of invariant positive linear functionals on $L^{ ext{infty}}( ext{R}_+)$.

Connection between invariance properties and linear mappings on function spaces.

Applications to summability methods in analysis.

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.