On the Riesz Almost Convergent Sequences Space

Abstract

The purpose of this paper is to introduce new spaces $\widehat{f}$ and ${\widehat{f}}_0$ that consist of all sequences whose Riesz transforms of order one are in the spaces $f$ and $f_0$, respectively. We also show that $\widehat{f}$ and ${\widehat{f}}_0$ are linearly isomorphic to the spaces $f$ and $f_0$, respectively. The $\beta \text{-}$ and $\gamma \text{-}$duals of the spaces $\widehat{f}$ and ${\widehat{f}}_0$ are computed. Furthermore, the classes $(\widehat{f}:\mu )$ and $(\mu :\widehat{f})$ of infinite matrices are characterized for any given sequence space $\mu$ and determine the necessary and sufficient conditions on a matrix $A$ to satisfy $B_R-\text{c}\text{o}\text{r}\text{e}\left(Ax\right)\subseteq K-\text{c}\text{o}\text{r}\text{e}\left(x\right)$, $B_R-\text{c}\text{o}\text{r}\text{e}\left(Ax\right)\subseteq st-\text{c}\text{o}\text{r}\text{e}\left(x\right)$ for all $x\in {\mathcal{l}}_{\infty }$.