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On the Riesz almost convergent sequences space. (English) Zbl 1250.46005
Summary: 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 $\beta$- and $\gamma$-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 $B_R - \text{core}(Ax) \subseteq K - \text{core}(x)$, $B_R - \text{core}(A_R) \subseteq st - \text{core}(x)$ for all $x \in \ell_\infty$.

46A45Sequence spaces
40C05Matrix methods in summability
Full Text: DOI
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