Ahmad, Z. U.; Mursaleen Köthe-Toeplitz duals of some new sequence spaces and their matrix maps. (English) Zbl 0647.46006 Publ. Inst. Math., Nouv. Sér. 42(56), 57-61 (1987). For a sequence \(x=(x_ n)\), define \(\Delta x=(x_ k-x_{k+1})\). Using this \(\Delta\) x, Kizmaz defined the sequence spaces \(\ell_{\infty}(\Delta)\), c(\(\Delta)\) and \(c_ 0(\Delta)\) as follows: \[ \ell_{\infty}(\Delta)=\{x=(x_ k)| \quad \Delta x\in \ell_{\infty}\}, \]\[ c(\Delta)=\{x=(x_ k)| \quad \Delta x\in c\}, \]\[ c_ 0(\Delta)=\{x=(x_ k)| \quad \Delta x\in c_ 0\}. \] 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 \(p=(p_ k)\) of strictly positive numbers. For example if c(p) is the Maddox sequence space of convergent sequences, the author considers \(\Delta c(p)=\{x=(x_ k)|\Delta\) \(x\in c(p)\}.\) Introducing the spaces \(\ell_{\infty}(p)\), c(p) and \(c_ 0(p)\), the author finds the first and second Köthe-Toeplitz duals of \(\Delta \ell_{\infty}(p)\) and asserts \(\Delta \ell_{\infty}(p)\) is perfect if and only if \(p\in \ell_{\infty}\). The necessary and sufficient conditions for an infinite matrix to transform \(\ell_ p\) to c(\(\Delta)\), \(\Delta \ell_{\infty}(p)\) to \(\ell_{\infty}\) and \(\Delta \ell_{\infty}(p)\) to c are obtained. Reviewer: D.Somasundaram Cited in 1 ReviewCited in 34 Documents MSC: 46A45 Sequence spaces (including Köthe sequence spaces) 40C05 Matrix methods for summability 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) Keywords:matrix transformations; Maddox sequence space; first and second Köthe- Toeplitz duals; perfect; infinite matrix PDFBibTeX XMLCite \textit{Z. U. Ahmad} and \textit{Mursaleen}, Publ. Inst. Math., Nouv. Sér. 42(56), 57--61 (1987; Zbl 0647.46006) Full Text: EuDML