On the Domain of Riesz Mean in the Space $\mathcal L_s^*$

Medine Yeşilkayagil, Feyzi Başar

Let $0<s<\infty$. In this study, we introduce the double sequence space $R^{qt}(\mathcal L_s)$ as the domain of four dimensional Riesz mean $R^{qt}$ in the space $\mathcal L_s$ of absolutely $s$-summable double sequences. Furthermore, we show that $R^{qt}(\mathcal L_s)$ is a Banach space and a barrelled space for $1<s<\infty$ and is not a barrelled space for $0<s<1$. We determine the $\alpha$- and $\beta(\vartheta)$-duals of the space $\mathcal L_s$ for $0<s<1$ and $\beta(bp)$-dual of the space $R^{qt}(\mathcal L_s)$ for $1<s<\infty$ where $\vartheta\in\{p,bp,r\}$. Finally, we characterize the classes $(\mathcal L_s:\mathcal M_u)$, $(\mathcal L_s:\mathcal C_{bp})$, $(R^{qt}(\mathcal L_s):\mathcal M_u)$ and $(R^{qt}(\mathcal L_s):\mathcal C_{bp})$ of four dimensional matrices in the cases both $0<s<1$ and $1<s<\infty$ 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.