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Zbl 1085.46002
Aydin, Cafer; Başar, Feyzi
On the new sequence spaces which include the spaces $c_0$ and $c$. (English)
[J] Hokkaido Math. J. 33, No. 2, 383-398 (2004). ISSN 0385-4035

Let $w$ denotes the set of all real sequences and consider for any $r\in (0,1)$ the matrix $A^r:= (a^{(r)}_{nk})$ with $$a^{(r)}_{nk}:= \cases {1+ r^k\over n+1},\quad & 0\le k\le n,\\ 0,\quad & k> n.\endcases$$ Consider the sets of sequences $$\align a^{(r)}_0 &:= \Biggl\{x= (x_k)\in w\,\Biggl\vert\,\lim_n {1\over n+1} \sum^n_{k=0} (1+ r^k)= 0\Biggr\},\\ a^{(r)}_c &:= \Biggl\{x= (x_k)\in w\,\Biggl\vert\,\lim_n{1\over n+1} \sum^n_{k=0} (1+ r^k)\in \bbfR\Biggr\}\endalign$$ which are the spaces of sequences the $A^r$-transform of which belongs to $c_0$ and to $c$, respectively. In the present paper the authors, following similar work done by Wang (1978), Ng and Lee (1978), Malkowsky (1997) and, recently, by Altay and Başar (2002), introduce the above spaces, they prove that they are linearly isomorphic to $c_0$ and to $c$, respectively, construct their bases and study the $\alpha$-, $\beta$- and $\gamma$-duals of them. Finally, they characterize certain classes of matrix transformations involving the spaces $a^{(r)}_0$ and $a^{(r)}_c$, e.g., the classes $(a^{(r)}_0, \ell_p)$, $(a^{(r)}_c, c)$ and others. The authors seem unaware of a recent paper by {\it E.~Malkowsky} [Rend. Circ. Mat. Palermo (2) 68, 641--655 (2002; Zbl 1028.46015)] by certain results of which one could obtain Schauder bases for $a^{(r)}_0$ and $a^{(r)}_c$, the $\alpha$-, $\beta$-, $\gamma$-duals for $a^{(r)}_0$ and $A^{(r)}_c$ as well as the classes $(a^{(r)}_0, \ell_p)$, $(a^{(r)}_c, c)$ as special cases.
[Constantine G. Lascarides (Athens)]

MSC 2000:: *46A45 Sequence spaces
46B45 Banach sequence spaces
46A35 Summability and bases in topological linear spaces

Keywords: sequence spaces; duals and basis of a sequence space; matrix transformations

Citations: Zbl 0415.46009; Zbl 0408.46012; Zbl 0942.40006; Zbl 1058.46002; Zbl 1028.46015

Cited in: Zbl 1199.46017 Zbl 1132.46005 Zbl 1072.46007

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