Four-dimensional generalized difference matrix and some double sequence spaces

Orhan Tuǧ

doi:10.1186/s13660-017-1423-y

Abstract

In this study, I introduce some new double sequence spaces B(mathcal {M}_{u}), B(mathcal{C}_{p}), B(mathcal{C}_{bp}), B(mathcal {C}_{r}) and B(mathcal{L}_{q}) as the domain of four-dimensional generalized difference matrix B(r,s,t,u) in the spaces mathcal {M}_{u}, mathcal{C}_{p}, mathcal{C}_{bp}, mathcal{C}_{r} and mathcal{L}_{q}, respectively. I show that the double sequence spaces B(mathcal{M}_{u}), B(mathcal{C}_{bp}) and 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}) and B(mathcal{C}_{bp}), the beta(vartheta)-duals of the spaces B(M_{u}), B(mathcal{C}_{p}), B(mathcal{C}_{bp}), B(mathcal{C}_{r}) and B(mathcal{L}_{q}), where varthetain{p,bp,r}, and the γ-dual of the spaces B(mathcal{M}_{u}), B(mathcal{C}_{bp}) and B(mathcal{L}_{q}). Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces B(mathcal{M}_{u}), B(mathcal{C}_{p}), B(mathcal{C}_{bp}), B(mathcal{C}_{r}) and B(mathcal{L}_{q}) of double sequences.

Highlights

We denote the set of all complex-valued double sequences by which is a vector space with coordinatewise addition and scalar multiplication
Altay and Başar [ ] have recently studied the double series spaces BS, BS(t), CSθ and BV whose sequences of partial sums are in the spaces Mu, Mu(t), Cθ and Lu, respectively, where θ ∈ {p, bp, r}
They studied some topological properties of those spaces and computed the α-duals of the spaces BS, CSbp and BV and the β(θ)-duals of the spaces CSbp and CSr of double series

Summary

Introduction

We denote the set of all complex-valued double sequences by which is a vector space with coordinatewise addition and scalar multiplication. I characterize the classes of four-dimensional matrix mappings defined on the spaces B(Mu), B(Cp), B(Cbp), B(Cr) and B(Lq) of double sequences. Let λ and μ be two double sequence spaces and A = (amnkl) be any four-dimensional complex infinite matrix. Let y = (ykl) ∈ Mu and define x = (xmn) via the sequence y by relation

Results

Conclusion