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.