Dilation invariant Banach limits

TL;DR

This paper investigates two subclasses of Banach limits invariant under specific operators, establishing their hierarchy and their maximal distance from extreme points, advancing understanding of Banach limit invariance properties.

Contribution

It introduces and compares two classes of invariant Banach limits, proving one is a proper subset of the other and analyzing their geometric properties.

Findings

The class of Cesàro-invariant Banach limits is a proper subset of the dilation-invariant class.

Both classes are maximally distant from the set of extreme Banach limits.

The study clarifies the structure and invariance properties of Banach limits.

Abstract

We study two subclasses of Banach limits: the one consisting of Banach limits which are invariant with respect of the Ces\`aro operator and another one consists of Banach limits which are invariant with respect to all dilations. We prove that the first is a proper subset of the second. We also show that these classes are at the maximal distance from the set ${\rm ext} \mathfrak B$ of all extreme points of the set of all Banach limits.