Invariant Banach limits and applications

By E. Semenov, Fedor Sukochev. 2010

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Let l∞ be the space of all bounded sequences x=(x1,x2,…) with the norm ‖x‖l∞=supn|xn| and let L(l∞) be the set of all bounded linear operators on l∞. We present a set of easily verifiable sufficient conditions on an operator H∈L(l∞), guaranteeing the existence of a Banach limit B on l∞ such that B=BH. We apply our results to the classical Cesaro operator C on l∞ and give necessary and sufficient condition for an element x∈l∞ to have fixed value Bx for all Cesaro 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 Cesaro-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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f0be1af7ca9026993b8254d0de50e63f68857dc2