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

Abstract

In this study, we define the spaces M_u(Δ), C_p(Δ), C_0p(Δ), C_r(Δ) and ℒ_q(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 M_u(Δ) and the β(v)-dual of the space C_η(Δ) of double sequences, where v, η ∈ {p,bp,r}. Finally, we characterize the classes (μ : C_v(Δ)) for v ∈ {p,bp,r} of four dimensional matrix transformations, where μ is any given space of double sequences.