Generalized limits and sequence of matrices

Özgüç I., Taş E., YURDAKADİM T.

Positivity, vol.24, no.3, pp.553-563, 2020 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 24 Issue: 3
  • Publication Date: 2020
  • Doi Number: 10.1007/s11117-019-00696-y
  • Journal Name: Positivity
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, ABI/INFORM, Business Source Elite, Business Source Premier, MathSciNet, zbMATH
  • Page Numbers: pp.553-563
  • Keywords: B-statistical limit superior and inferior, Banach limit, Sequence of infinite matrices, The Hahn–Banach extension theorem
  • Bilecik Şeyh Edebali University Affiliated: Yes

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(χK) = 0 for every characteristic sequence χK of sets, K, of natural density zero. Recently the comparison of such functionals and some applications have been examined. In this paper we define SB -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 SA-limits and A-Banach limits. We also show that the sets of all SB -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.