Let lΣ be the space of all bounded sequences x=(x1,x2,...) with the norm Let lΣ be the spaceLet lΣ be the spacexLet lΣ be the spaceLet lΣ be the spaceLet lΣ be the space=supn|xn| and let L(Let lΣ be the spaceLet lΣ be the spaceLet lΣ be the space) be the set of all bounded linear operators on lΣ. We present a set of easily verifiable sufficient conditions on an operator H lΣL(), guaranteeing the existence of a Banach limit B on lΣ such that B=BH. We apply our results to the classical Cesàro operator C on lΣ and give necessary and sufficient condition for an element lΣ 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.
展开▼
