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Matrix transformations in spaces of bounded and convergent difference sequences of order \(m\). (English) Zbl 0872.40002

Summary: Let \(m\) be a positive integer. We define the sets \(c_0(\Delta^{(m)})\), \(c(\Delta^{(m)})\) and \(l_\infty(\Delta^{(m)})\) of sequences the \(m\)th order differences of which are convergent to zero, convergent or bounded, give Schauder bases for \(c_0(\Delta^{(m)})\) and \(c(\Delta^{(m)})\), and compute the \(\alpha\)- and \(\beta\)-duals of \(c_0(\Delta^{(m)})\), \(c(\Delta^{(m)})\) and \(l_\infty(\Delta^{(m)})\). Finally we characterize matrix transformations between these spaces. For \(m=1\), some of our results yield those in H. Kizmaz [Can. Math. Bull. 24, No. 2, 169-176 (1981; Zbl 0454.46010)].

MSC:

40H05 Functional analytic methods in summability
46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0454.46010
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