Invariant Banach limits and applications

Let ∞ be the space of all bounded sequences x = (x1, x2, . . .) with the norm x ∞ = sup n |xn| and let L( ∞) be the set of all bounded linear operators on ∞. We present a set of easily verifiable sufficient conditions on an operator H ∈ L( ∞), guaranteeing the existence of a Banach limit B on ∞ such that B = BH. We apply our results to the classical Cesàro operator C on ∞ and give necessary and sufficient condition for an element x ∈ ∞ 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. © 2010 Elsevier Inc. All rights reserved.