Four-Dimensional Generalized Difference Matrix and Almost Convergent Double Sequence Spaces

Abstract

Tuǧ and Başar [3] have recently studied the concept of four dimensional generalized difference matrix B(r, s, t, u) and its matrix domain in some double sequence spaces. In this present paper, as a natural continuation of [3], we introduce new almost null and almost convergent double sequence spaces $B(C_f)$ and $B(C_{f_0})$ as the domain of four-dimensional generalized difference matrix B(r, s, t, u) in the spaces $C_f$ and $C_{f_0}$, respectively. Firstly, we prove that the spaces $B(C_f)$ and $B(C_{f_0})$ of double sequences are Banach spaces under some certain conditions. We give some inclusion relations with some topological properties. Moreover, we determine the $\alpha -$ dual , $\beta (bp)-$dual and $\gamma -$ dual of the spaces $B(C_f)$. Finally, we characterize the classes of four dimensional matrix mappings defined on the spaces $B(C_f)$ of double sequences.