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Köthe-Toeplitz duals of some new sequence spaces and their matrix maps. (English) Zbl 0647.46006
For a sequence $x=(x\sb n)$, define $\Delta x=(x\sb k-x\sb{k+1})$. Using this $\Delta$ x, Kizmaz defined the sequence spaces $\ell\sb{\infty}(\Delta)$, c($\Delta)$ and $c\sb 0(\Delta)$ as follows: $$\ell\sb{\infty}(\Delta)=\{x=(x\sb k)\vert \quad \Delta x\in \ell\sb{\infty}\},$$ $$c(\Delta)=\{x=(x\sb k)\vert \quad \Delta x\in c\},$$ $$c\sb 0(\Delta)=\{x=(x\sb k)\vert \quad \Delta x\in c\sb 0\}.$$ If E is any one of the above spaces, we have $E\subset \Delta E$. The aim of the present paper is to extend the above sequence spaces to the sequence spaces of Maddox and Simons by considering a sequence $p=(p\sb k)$ of strictly positive numbers. For example if c(p) is the Maddox sequence space of convergent sequences, the author considers $\Delta c(p)=\{x=(x\sb k)\vert\Delta$ $x\in c(p)\}.$ Introducing the spaces $\ell\sb{\infty}(p)$, c(p) and $c\sb 0(p)$, the author finds the first and second Köthe-Toeplitz duals of $\Delta \ell\sb{\infty}(p)$ and asserts $\Delta \ell\sb{\infty}(p)$ is perfect if and only if $p\in \ell\sb{\infty}$. The necessary and sufficient conditions for an infinite matrix to transform $\ell\sb p$ to c($\Delta)$, $\Delta \ell\sb{\infty}(p)$ to $\ell\sb{\infty}$ and $\Delta \ell\sb{\infty}(p)$ to c are obtained.
Reviewer: D.Somasundaram

##### MSC:
 46A45 Sequence spaces 40C05 Matrix methods in summability 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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