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

TL;DR

This paper introduces new strongly almost convergent double sequence spaces based on generalized difference matrices, proves their Banach space structure, and characterizes related matrix classes and duals.

Contribution

It defines and analyzes new strongly almost null and convergent double sequence spaces as domains of generalized difference matrices, extending existing sequence space theory.

Findings

The spaces $B[\\mathcal{C}_f]$ and $B[\mathcal{C}_{f_0}]$ are Banach spaces.

Includes inclusion relations among new and existing sequence spaces.

Calculates dual spaces and characterizes matrix classes related to these 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.