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Generalized spaces of difference sequences. (English) Zbl 0873.46014

Abstract of the paper: “Let \(\ell_\infty\), \(c\) and \(c_0\) be the Banach spaces of bounded, convergent and null sequences \(x=(x_k)^\infty_1\), respectively. Write \(\Delta x=(x_k- x_{k+1})^\infty_1\) and \(\Delta^2x=(\Delta x_k-\Delta x_{k+1})^\infty_1\). In [Canad. Math. Bull. 24, 169-176 (1981; Zbl 0454.46010)], H. Kizmaz has introduced and studied the sequence spaces, \(E(\Delta)=\{x:\Delta x\in E\}\), where \(E\in\{c_0,c,\ell_\infty\}\). Recently, [Turk. J. Math. 17, No. 1, 18-24 (1993; Zbl 0826.40001)], Mikail Et defined the sets \(E(\Delta^2)= \{x:\Delta^2 x\in E\}\). He obtained \(\alpha\)-duals of these sets and characterized the matrix class \((E,F(\Delta^2))\), where \(E,F\in\{c_0,c,\ell_\infty\}\). In this paper, we generalize these sets and define \(E(u;\Delta^2)= \{x: u\cdot\Delta^2x\in E\}\), where \(u=(u_k)\) is another sequence such that \(u_k\neq 0\) \((k=1,2,\dots)\). We obtain \(\alpha\)- and \(\beta\)-duals of these sets and further we characterize the matrix classes \((E(u;\Delta^2),F)\) and \((E,F(u;\Delta^2))\)”.

MSC:

46B45 Banach sequence spaces
46A45 Sequence spaces (including Köthe sequence spaces)
40H05 Functional analytic methods in summability
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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