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On some new generalized difference sequence spaces. (English) Zbl 0922.40003

Let \(\omega\) be the space of all complex sequences \(x=(x_k)\). The authors introduce certain generalized difference sequence spaces related to \(\ell_{\infty}\), \(c_0\), \(c\), i.e. to the Banach spaces of bounded, null and convergent sequences, respectively, with the aid of the linear operator \(\Delta^r\), where \(\Delta^r x=(\Delta^r x_k)=(\Delta^{r-1} x_k-\Delta^{r-1} x_{k+1})\), see I. J. Maddox [Math. Proc. Camb. Philos. Soc. 85, 345-350 (1979; Zbl 0417.40007)] and S. Nanda [Bull. Calcutta Math. Soc. 76, 236-240 (1984; Zbl 0588.40009)]. They investigate inclusion relations and Köthe-Toeplitz duals. Here the dual of \(X\) is defined by \(X^{\alpha}=\{a=(a_k)\in \omega\mid \sum_k | a_kx_k| <\infty\) for every \(x\in X\}\).

MSC:

40A05 Convergence and divergence of series and sequences
40C05 Matrix methods for summability
46A45 Sequence spaces (including Köthe sequence spaces)
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