On the new sequence spaces which include the spaces \(c_0\) and \(c\)
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 \[ a^{(r)}_{nk}:= \begin{cases} {1+ r^k\over n+1},\quad & 0\leq k\leq n,\\ 0,\quad & k> n.\end{cases} \] Consider the sets of sequences \[ \begin{aligned} a^{(r)}_0 &:= \Biggl\{x= (x_k)\in w\,\Biggl|\,\lim_n {1\over n+1} \sum^n_{k=0} (1+ r^k)= 0\Biggr\},\\ a^{(r)}_c &:= \Biggl\{x= (x_k)\in w\,\Biggl|\,\lim_n{1\over n+1} \sum^n_{k=0} (1+ r^k)\in \mathbb{R}\Biggr\}\end{aligned} \] 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 \textit{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.
scientific article; zbMATH DE number 2158043
scientific article