Domain of difference matrix of order one in some spaces of double sequences

Abstract

In this study, we define the spaces Mu(Δ), Cp(Δ), C0p(Δ), Cr(Δ) and ℒq(Δ) of double sequences whose difference transforms are bounded , convergent in the Pringsheim's sense, null in the Pringsheim's sense, both convergent in the Pringsheim's sense and bounded, regularly convergent and absolutely q-summable, respectively, and also examine some inclusion relations related to those sequence spaces. Furthermore, we show that these sequence spaces are Banach spaces. We determine the alpha-dual of the space Mu(Δ) and the β(v)-dual of the space Cη(Δ) of double sequences, where v, η ∈ {p,bp,r}. Finally, we characterize the classes (μ : Cv(Δ)) for v ∈ {p,bp,r} of four dimensional matrix transformations, where μ is any given space of double sequences.