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On the domain of Riesz mean in the space Ls

Yeşilkayagil Medine (Uşak University, Department of Mathematics, Eylül Campus, Uşak, Turkey)
Başar Feyzi (İstanbul, Turkey)

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 < 1 and is not a barrelled space for 0 < s < 1. We determine the α- and β(ν)-duals of the space Ls for 0 < s ≤ 1 and β(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ν  {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 < 1 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.

Keywords: summability theory, double sequences, double series, double sequence spaces, alpha-, beta- and gamma-duals, 4-dimensional matrices and matrix transformations