On the Domain of Riesz Mean in the Space Ls *

Abstract

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 and establish that the inclusion
$R^{qt}(\mathcal{L}_{s})\subset R^{qt}(\mathcal{L}_{r})$ strictly
holds with $1<s<r<\infty$. Furthermore, we show that
$R^{qt}(\mathcal{L}_{s})$ is a Banach space and a barrelled space
for $1\leq s<\infty$ and is not a barrelled space for $0<s<1$. We
determine the $\alpha$-, $\gamma$- and $\beta(\vartheta)$-duals of
the spaces $\mathcal{L}_{s}$ and $\alpha$ dual of
$R^{qt}(\mathcal{L}_{s})$, $(0<s\leq 1)$. 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 $1\leq s<\infty$ 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.