Dilation invariant Banach limits

Abstract

We study two subclasses of Banach limits: the one consisting of Banach limits which are invariant with respect of the Cesàro 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 $\mathrm{ext}\mathfrak{B}$ of all extreme points of the set of all Banach limits.

10 pages