Invariant Banach limits

The talk is devoted to: properties of the space of almost convergent sequences and criteria for belonging a sequence to this space; special asymptotic characteristics of bounded sequences; invariant Banach limits and new classes of linear operators defined using them; properties of sets defined using Sacheston functionals. A Banach limit is a continuous, positive, normalized, shift-invariant linear functional on the space of bounded sequences that coincides with the usual limit on any convergent sequence.
A bounded sequence is called almost convergent if the value of the Banach limit on it does not depend on the choice of this Banach limit. The upper and lower (nonlinear) Sacheton functionals are called the maximum and minimum values that all Banach limits can take on a given sequence. A set of bounded sequences is called separating if, for any two different Banach limits, there is at least one element in it on which these Banach limits take different values. A natural reinforcement of the requirement of invariance with respect to shear is the requirement of invariance with respect to some other operators (the stretching operator, the Cesaro operator), which leads to the concept of invariant Banach limits.