Domain of difference matrix of order one in some spaces of double sequences

TL;DR

This paper introduces new double sequence spaces based on difference transformations, explores their properties including duals, and characterizes matrix transformations between these spaces, expanding the theoretical framework of sequence space analysis.

Contribution

It defines several new double sequence spaces related to difference transformations, proves they are Banach spaces, and characterizes their duals and matrix transformations, which is a novel extension in the field.

Findings

Spaces are Banach spaces.

Determined alpha-dual and beta(v)-dual of specific spaces.

Characterized classes of matrix transformations.

Abstract

In this study, we define the spaces $\mathcal{M}_{u}(\Delta),\mathcal{C}_{p}(\Delta),\mathcal{C}_{0p}(\Delta), \mathcal{C}_{r}(\Delta)$ and $\mathcal{L}_{q}(\Delta)$ of double sequences whose difference transforms are bounded , convergent in the Pringsheim's sense, null in the Pringsheim's sense, both convergent in the Pringsheim's sense and bounded, regularly convergent and absolutely $q-$summable, respectively, and also examine some inclusion relations related to those sequence spaces. Furthermore, we show that these sequence spaces are Banach spaces . We determine the alpha-dual of the space $\mathcal{M}_{u}(\Delta)$ and the $\beta(v)-$dual of the space $\mathcal{C}_{\eta}(\Delta)$ of double sequences, where $v,\eta\in \{p,bp,r\}$. Finally, we characterize the classes $(\mu:\mathcal{C}_{v}(\Delta))$ for $v\in \{p,bp,r\}$ of four dimensional matrix transformations, where $\mu$ is any given space of double sequences.