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 $$S_{{\mathfrak {B}}}$$ -limits and $${\mathfrak {B}}$$-Banach limits where $${\mathfrak {B}}$$ is a sequence of infinite matrices. It is clear that if $$\mathfrak {B=(}A\mathfrak {)}$$ then these definitions reduce to $$S_{A}$$-limits and A-Banach limits. We also show that the sets of all $$ S_{{\mathfrak {B}}}$$ -limits and Banach limits are distinct but their intersection is not empty. Furthermore, we obtain that the generalized limits generated by $${\mathfrak {B}}$$ where $${\mathfrak {B}}$$ is strongly regular is equal to the set of Banach limits.