On the new sequence spaces which include the spaces \(c_0\) and \(c\)
type: Item
instance of: scholarly article
title: On the new sequence spaces which include the spaces \(c_0\) and \(c\)
zbMATH Open document ID: 1085.46002
author: C. Aydin
author: Feyzi Başar
published in: Hokkaido Mathematical Journal
publication date: 2005-04-18T00:00:00Z
review text: 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.
reviewed by: Constantine G. Lascarides
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46B45
Mathematics Subject Classification ID: 46A35
zbMATH DE Number: 2158043
zbMATH Keywords: sequence spaces
zbMATH Keywords: duals and basis of a sequence space
zbMATH Keywords: matrix transformations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1285766172
OpenAlex ID: W2044847736
DOI: 10.14492/HOKMJ/1285766172
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description: scientific article; zbMATH DE number 2158043