Invariant Banach limits and applications

Let ℓ∞ℓ∞ be the space of all bounded sequences x=(x1,x2,…)x=(x1,x2,…) with the norm‖x‖ℓ∞=supn|xn| and let L(ℓ∞)L(ℓ∞) be the set of all bounded linear operators on ℓ∞ℓ∞. We present a set of easily verifiable sufficient conditions on an operator H∈L(ℓ∞)H∈L(ℓ∞), guaranteeing the existence of a Banach limit B   on ℓ∞ℓ∞ such that B=BHB=BH. We apply our results to the classical Cesàro operator C   on ℓ∞ℓ∞ and give necessary and sufficient condition for an element x∈ℓ∞x∈ℓ∞ to have fixed value Bx for all Cesàro invariant Banach limits B. Finally, we apply the preceding description to obtain a characterization of “measurable elements” from the (Dixmier–)Macaev–Sargent ideal of compact operators with respect to an important subclass of Dixmier traces generated by all Cesàro-invariant Banach limits. It is shown that this class is strictly larger than the class of all “measurable elements” with respect to the class of all Dixmier traces.