On the Generalized Difference Matrix Domain on Strongly Almost Convergent Double Sequence Spaces

Abstract

Most recently, some new double sequence spaces $B(\mathcal{M}_{u})$, $B(\mathcal{C}_{\vartheta})$ where $\vartheta=\{b,bp,r,f,f_0\}$ and $B(\mathcal{L}_{q})$ for $0<q<\infty$ have been introduced as four-dimensional generalized difference matrix $B(r,s,t,u)$ domain on the double sequence spaces $\mathcal{M}_{u}$, $\mathcal{C}_{\vartheta}$ where $\vartheta=\{b,bp,r,f,f_0\}$ and $\mathcal{L}_{q}$ for $0<q<\infty$, and some topological properties, dual spaces, some new four-dimensional matrix classes and matrix transformations related to these spaces have also been studied by Tu\u{g} and Ba\c{s}ar and Tu\u{g} (see \cite{OT,OT2,Orhan,Orhan 2}). In this present paper, we introduce new strongly almost null and strongly almost convergent double sequence spaces $B[\mathcal{C}_f]$ and $B[\mathcal{C}_{f_0}]$ as domain of four-dimensional generalized difference matrix $B(r,s,t,u)$ in the spaces $[\mathcal{C}_f]$ and $[\mathcal{C}_{f_0}]$, respectively. Firstly, we prove that the new double sequence spaces $B[\mathcal{C}_f]$ and $B[\mathcal{C}_{f_0}]$ are Banach spaces with its norm. Then, we give some inclusion relations including newly defined strongly almost convergent double sequence spaces. Moreover, we calculate the $\alpha-$dual, $\beta(bp)-$dual and $\gamma-$dual of the space $B[\mathcal{C}_f]$. Finally, we characterize new four-dimensional matrix classes $([\mathcal{C}_{f}];\mathcal{C}_{f})$, $([\mathcal{C}_{f}];\mathcal{M}_{u})$, $(B[\mathcal{C}_{f}];\mathcal{C}_{f})$, $(B[\mathcal{C}_{f}];\mathcal{M}_{u})$ and we complete this work with some significant results.