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Generalizations of Köthe-Toeplitz duals and null duals of new difference sequence spaces. (English) Zbl 1367.46005

J. Contemp. Math. Anal., Armen. Acad. Sci. 51, No. 3, 125-133 (2016) and Izv. Nats. Akad. Nauk Armen., Mat. 51, No. 3, 28-40 (2016).
Summary: The main purpose of the paper is to generalize the notions of the Köthe-Toeplitz duals and null duals of sequence spaces by introducing the concepts of \(\alpha EF\)-, \(\beta EF\)-, \(\gamma EF\)-duals and \(NEF\)-duals, where \(E = (E_n)\) and \(F = (F_n)\) are two partitions of finite subsets of the positive integers. These duals are computed for the classical sequence spaces \(l_c\) and \(c_0\). The other purpose of the paper is to introduce the sequence spaces \[ X(E,\Delta) = \left\{ x = \left(x_k\right):\left(\sum\limits_{i \in {E_k}} x_i - \sum\limits_{i \in {E_{k - 1}}} {{x_i}}\right)^\infty_{k=1}\in X \right\} \] where \(X \in \left\{l_\infty,c,c_0\right\}\). We investigate the topological properties of these spaces, establish some inclusion relations between them, and compute the \(NEF\)- (or null) duals for these spaces.

MSC:

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

[1] F. Başar, Summability Theory and Its Applications (Bentham, Istanbul, 2012). · Zbl 1342.40001
[2] F. Başar, B. Altay and M. Mursaleen, “Some generalizations of the space bvp of p-bounded variation sequences”, Nonlinear Anal., 68 2, 273-287, 2008. · Zbl 1132.46002 · doi:10.1016/j.na.2006.10.047
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