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

TL;DR

This paper introduces new double sequence spaces based on the four-dimensional forward difference matrix, explores their properties, duals, and matrix class characterizations in the context of advanced sequence space theory.

Contribution

It defines novel double sequence spaces using the four-dimensional forward difference matrix and investigates their topological, algebraic properties, duals, and matrix class characterizations.

Findings

New double sequence spaces $ ext{\mathcal{M}_u(\Delta)}$ and $ ext{\mathcal{C}_\vartheta(\Delta)}$ introduced.

Determined the duals of the new sequence spaces.

Characterized matrix classes involving these new 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 $\mu=\{\mathcal{M}_u,\mathcal{C}_{\vartheta}\}$.