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Some new difference sequence spaces. (English) Zbl 1072.46007
Summary: The difference sequence spaces (Δ), c(Δ) and c 0 (Δ) were studied by H. Kizmaz [Can. Math. Bull. 24, 169–176 (1981; Zbl 0454.46010)]. The sequence spaces a 0 r and a c r have been recently defined and examined by C. Aydin and F. Basar [Hokkaido Math. J. 33, No. 2, 383–398 (2004; Zbl 1085.46002)]. The main purpose of the present paper is to introduce the spaces a 0 r (Δ) and a c r (Δ) of difference sequences. Moreover, it is proven that the spaces a 0 r (Δ) and a c r (Δ) are BK-spaces including the spaces c 0 and c, and some inclusion relations are given. It is also proven that the sequence space a 0 r has the AD property while the space a 0 r (Δ) does not. Furthermore, a basis and the α-, β- and γ-duals of the spaces a 0 r (Δ) and a c r (Δ) are determined. The last section of the paper is devoted to characterizations of the matrix classes (a c r (Δ): p ) and (a c r (Δ):c), and the characterizations of some other matrix classes are obtained by means of a given basic lemma, where 1p.

46A45Sequence spaces