Some new difference sequence spaces

摘要

The difference sequence spaces ℓ∞(Δ),c(Δ) and c0(Δ) were studied by Kızmaz [Canad. Math. Bull. 24 (2) (1981) 169]. The sequence spaces a0r and acr have been recently defined and examined by Aydın and Başar in [Hokkaido Math. J., in press]. The main purpose of the present paper is to introduce the spaces ar0(Δ) and arc(Δ) of difference sequences. Moreover, it is proved that the spaces ar0(Δ) and arc(Δ) are the BK-spaces including the spaces c0 and c, and some inclusion relations have been given. It is also proved that the sequence space a0r has AD property while the space a0r(Δ) has not. Furthermore, the basis and the α-, β- and γ-duals of the spaces ar0(Δ) and arc(Δ) have been determined. The last section of the paper has been devoted to theorems on the characterizations of the matrix classes (arc(Δ):ℓp) and (arc(Δ):c), and the characterizations of some other matrix classes have been obtained by means of a given basic lemma, where 1⩽p⩽∞.