Four-dimensional generalized difference matrix and some double sequence spaces

Abstract

Abstract In this study, I introduce some new double sequence spaces B ( M u ) $B(\mathcal {M}_{u})$ , B ( C p ) $B(\mathcal{C}_{p})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ , B ( C r ) $B(\mathcal {C}_{r})$ and B ( L q ) $B(\mathcal{L}_{q})$ as the domain of four-dimensional generalized difference matrix B ( r , s , t , u ) $B(r,s,t,u)$ in the spaces M u $\mathcal {M}_{u}$ , C p $\mathcal{C}_{p}$ , C b p $\mathcal{C}_{bp}$ , C r $\mathcal{C}_{r}$ and L q $\mathcal{L}_{q}$ , respectively. I show that the double sequence spaces B ( M u ) $B(\mathcal{M}_{u})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ and B ( C r ) $B(\mathcal{C}_{r})$ are the Banach spaces under some certain conditions. I give some inclusion relations with some topological properties. Moreover, I determine the α-dual of the spaces B ( M u ) $B(M_{u})$ and B ( C b p ) $B(\mathcal{C}_{bp})$ , the β ( ϑ ) $\beta(\vartheta)$ -duals of the spaces B ( M u ) $B(M_{u})$ , B ( C p ) $B(\mathcal{C}_{p})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ , B ( C r ) $B(\mathcal{C}_{r})$ and B ( L q ) $B(\mathcal{L}_{q})$ , where ϑ ∈ { p , b p , r } $\vartheta\in\{p,bp,r\}$ , and the γ-dual of the spaces B ( M u ) $B(\mathcal{M}_{u})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ and B ( L q ) $B(\mathcal{L}_{q})$ . Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces B ( M u ) $B(\mathcal{M}_{u})$ , B ( C p ) $B(\mathcal{C}_{p})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ , B ( C r ) $B(\mathcal{C}_{r})$ and B ( L q ) $B(\mathcal{L}_{q})$ of double sequences.

Keywords

• four-dimensional generalized difference matrix
• matrix domain
• double sequence spaces
• alpha-dual
• beta-dual
• gamma-dual