On the domain of riesz mean in the space Ls*

Yeşilkayagil M., Başar F.

Filomat, vol.31, no.4, pp.925-940, 2017 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 31 Issue: 4
  • Publication Date: 2017
  • Doi Number: 10.2298/fil1704925y
  • Journal Name: Filomat
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.925-940
  • Keywords: 4-dimensional matrices and matrix transformations, Alpha-, beta- and gamma-duals, Double sequence spaces, Double sequences, Double series, Summability theory
  • Uşak University Affiliated: Yes

Abstract

Let 0 < s < ∞. In this study, we introduce the double sequence space Rqt (Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt (Ls) is a Banach space and a barrelled space for 1 ≤ s < ∞ and is not a barrelled space for 0 < s < 1. We determine the α- and β(ϑ)-duals of the space Ls for 0 < s ≤ 1 and β(bp)-dual of the space Rqt (Ls) for 1 < s < ∞, where ϑ ∈ {p, bp, r}. Finally, we characterize the classes (Ls: Mu), (Ls: Cbp), (Rqt (Ls): Mu) and (Rqt (Ls): Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ≤ s < ∞ together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.