Generalized limits and sequence of matrices

Abstract

Banach has proved that there exist positive linear regular functionals on m such that they are invariant under shift operator where m is the space of all bounded real sequences. It has also been shown that there exists positive linear regular functionals L on m such that L(chi(K))=0 for every characteristic sequence chi(K) of sets, K, of natural density zero. Recently the comparison of such functionals and some applications have been examined. In this paper we define S-B-limits and B-Banach limits where B is a sequence of infinite matrices. It is clear that if B=(A) then these definitions reduce to S-A-limits and A-Banach limits. We also show that the sets of all S-B-limits and Banach limits are distinct but their intersection is not empty. Furthermore, we obtain that the generalized limits generated by B where B is strongly regular is equal to the set of Banach limits.