Applications of Orlicz–Pettis theorem in vector valued multiplier spaces of generalized weighted mean fractional difference operators

• Main Authors: Kuldip Raj, Swati Jasrotia, M. Mursaleen
• Format: Article
• Language: English
• Published: SpringerOpen 2021-08-01
• Series: Advances in Difference Equations
• Subjects: Almost convergence; Generalized weighted mean operator G ( u , v ) $G(u,v)$; Weakly unconditionally Cauchy series; Unconditionally Cauchy series; Orlicz–Pettis theorem
• Online Access: https://doi.org/10.1186/s13662-021-03531-5

Abstract In this study, we deal with some new vector valued multiplier spaces S G h ( ∑ k z k ) $S_{G_{h}}(\sum_{k}z_{k})$ and S w G h ( ∑ k z k ) $S_{wG_{h}}(\sum_{k}z_{k})$ related with ∑ k z k $\sum_{k}z_{k}$ in a normed space Y. Further, we obtain the completeness of these spaces via weakly unconditionally Cauchy series in Y and Y ∗ $Y^{*}$ . Moreover, we show that if ∑ k z k $\sum_{k}z_{k}$ is unconditionally Cauchy in Y, then the multiplier spaces of G h $G_{h}$ -almost convergence and weakly G h − $G_{h}-$ almost convergence are identical. Finally, some applications of the Orlicz–Pettis theorem with the newly formed sequence spaces and unconditionally Cauchy series ∑ k z k $\sum_{k}z_{k}$ in Y are given.