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Some new difference sequence spaces. (English) Zbl 1072.46007
Summary: The difference sequence spaces ${\ell }_{\infty }\left({\Delta }\right)$, $c\left({\Delta }\right)$ and ${c}_{0}\left({\Delta }\right)$ 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 ${a}_{0}^{r}\left({\Delta }\right)$ and ${a}_{c}^{r}\left({\Delta }\right)$ of difference sequences. Moreover, it is proven that the spaces ${a}_{0}^{r}\left({\Delta }\right)$ and ${a}_{c}^{r}\left({\Delta }\right)$ 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 ${a}_{0}^{r}\left({\Delta }\right)$ does not. Furthermore, a basis and the $\alpha$-, $\beta$- and $\gamma$-duals of the spaces ${a}_{0}^{r}\left({\Delta }\right)$ and ${a}_{c}^{r}\left({\Delta }\right)$ are determined. The last section of the paper is devoted to characterizations of the matrix classes $\left({a}_{c}^{r}\left({\Delta }\right):{\ell }_{p}\right)$ and $\left({a}_{c}^{r}\left({\Delta }\right):c\right)$, and the characterizations of some other matrix classes are obtained by means of a given basic lemma, where $1\le p\le \infty$.

##### MSC:
 46A45 Sequence spaces