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

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.