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Some new difference sequence spaces. (English) Zbl 1072.46007
Summary: The difference sequence spaces $\ell_\infty(\Delta)$, $c (\Delta)$ and $c_0(\Delta)$ were studied by {\it H. Kizmaz} [Can. Math. Bull. 24, 169--176 (1981; Zbl 0454.46010)]. The sequence spaces $a^r_0$ and $a^r_c$ have been recently defined and examined by {\it C. Aydin} and {\it 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 $a^r_0(\Delta)$ and $a^r_c(\Delta)$ of difference sequences. Moreover, it is proven that the spaces $a^r_0 (\Delta)$ and $a^r_c(\Delta)$ are BK-spaces including the spaces $c_0$ and $c$, and some inclusion relations are given. It is also proven that the sequence space $a^r_0$ has the AD property while the space $a^r_0 (\Delta)$ does not. Furthermore, a basis and the $\alpha$-, $\beta$- and $\gamma$-duals of the spaces $a^r_0 (\Delta)$ and $a^r_c(\Delta)$ are determined. The last section of the paper is devoted to characterizations of the matrix classes $(a^r_c (\Delta):\ell_p)$ and $(a^r_c(\Delta):c)$, and the characterizations of some other matrix classes are obtained by means of a given basic lemma, where $1\le p\le\infty$.

46A45Sequence spaces
Full Text: DOI
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