Some new difference sequence spaces

The difference sequence spaces l(infinity)(Delta), c(Delta) and c(0)(Delta) were studied by Kizmaz [Canad. Math. Bull. 24 (2) (1981) 169]. The sequence spaces a(0)(r) and a(c)(r) have been recently defined and examined by Aydin and Basar in [Hokkaido Math. J., in press]. The main purpose of the present paper is to introduce the spaces a(0)(r)(Delta) and a(c)(r)(Delta) of difference sequences. Moreover, it is proved that the spaces a(0)(r)(Delta) and a(c)(r)(Delta) are the BK-spaces including the spaces c(0) and c, and some inclusion relations have been given. It is also proved that the sequence space a(0)(r) has AD property while the space a(0)(r)(Delta) has not. Furthermore, the basis and the alpha-, beta- and gamma-duals of the spaces a(0)(r)(Delta) and a(c)(r)(Delta) have been determined. The last section of the paper has been devoted to theorems on the characterizations of the matrix classes (a(c)(r)(Delta) : l(p)) and (a(c)(r)(Delta) : c), and the characterizations of some other matrix classes have been obtained by means of a given basic lemma, where 1 less than or equal to p less than or equal to infinity. (C) 2003 Elsevier Inc. All rights reserved.