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 $\mu={\mathcal{M}u,\mathcal{C}{\vartheta}}$.