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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 $x={\left({x}_{k}\right)}_{1}^{\infty }$, respectively. Write ${\Delta }x={\left({x}_{k}-{x}_{k+1}\right)}_{1}^{\infty }$ and ${{\Delta }}^{2}x={\left({\Delta }{x}_{k}-{\Delta }{x}_{k+1}\right)}_{1}^{\infty }$. In [Canad. Math. Bull. 24, 169-176 (1981; Zbl 0454.46010)], H. Kizmaz has introduced and studied the sequence spaces, $E\left({\Delta }\right)=\left\{x:{\Delta }x\in E\right\}$, where $E\in \left\{{c}_{0},c,{\ell }_{\infty }\right\}$. Recently, [Turk. J. Math. 17, No. 1, 18-24 (1993; Zbl 0826.40001)], Mikail Et defined the sets $E\left({{\Delta }}^{2}\right)=\left\{x:{{\Delta }}^{2}x\in E\right\}$. He obtained $\alpha$-duals of these sets and characterized the matrix class $\left(E,F\left({{\Delta }}^{2}\right)\right)$, where $E,F\in \left\{{c}_{0},c,{\ell }_{\infty }\right\}$. In this paper, we generalize these sets and define $E\left(u;{{\Delta }}^{2}\right)=\left\{x:u·{{\Delta }}^{2}x\in E\right\}$, where $u=\left({u}_{k}\right)$ is another sequence such that ${u}_{k}\ne 0$ $\left(k=1,2,\cdots \right)$. We obtain $\alpha$- and $\beta$-duals of these sets and further we characterize the matrix classes $\left(E\left(u;{{\Delta }}^{2}\right),F\right)$ and $\left(E,F\left(u;{{\Delta }}^{2}\right)\right)$”.

##### MSC:
 46B45 Banach sequence spaces 46A45 Sequence spaces 40H05 Functional analytic methods in summability 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)