On the Domain of Four-Dimensional Forward Difference Matrix in Some Double Sequence Spaces

Abstract

In this paper, we introduce some new double sequence spaces ${\mathcal{M}}_u(\Delta )$ and ${\mathcal{C}}_{\vartheta }(\Delta )$, where $\vartheta \in \{bp,bp0,r,r0\}$ as the domains of the four-dimensional forward difference matrix in the double sequence spaces ${\mathcal{M}}_u$ and ${\mathcal{C}}_{\vartheta }$, respectively. Then we investigate some topological and algebraic properties. Moreover, we determine the $\alpha -$, $\beta (\vartheta )-$, and $\gamma -$duals of the new spaces ${\mathcal{M}}_u(\Delta )$ and ${\mathcal{C}}_{\vartheta }(\Delta )$. Finally, we characterize four-dimensional matrix classes $(\lambda (\Delta ),\mu )$ and $(\mu ,\lambda (\Delta ))$, where $\lambda =\{{\mathcal{M}}_u,{\mathcal{C}}_{\vartheta }\}$ and $\mu =\{{\mathcal{M}}_u,{\mathcal{C}}_{\vartheta }\}$.

18 pages, 2 tables