Mursaleen Generalized spaces of difference sequences. (English) Zbl 0873.46014 J. Math. Anal. Appl. 203, No. 3, 738-745 (1996). 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 \((E,F(u;\Delta^2))\)”. Reviewer: C.G.Lascarides (Athens) Cited in 1 ReviewCited in 49 Documents 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.) Keywords:Banach spaces of bounded convergent and null sequences; matrix class Citations:Zbl 0454.46010; Zbl 0826.40001 PDFBibTeX XMLCite \textit{Mursaleen}, J. Math. Anal. Appl. 203, No. 3, 738--745 (1996; Zbl 0873.46014) Full Text: DOI