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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 n ), define Δx=(x k -x k+1 ). Using this Δ x, Kizmaz defined the sequence spaces (Δ), c(Δ) and c 0 (Δ) as follows:

(Δ)={x=(x k )|Δx },
c(Δ)={x=(x k )|Δxc},
c 0 (Δ)={x=(x k )|Δxc 0 }·

If E is any one of the above spaces, we have EΔE. 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 Δc(p)={x=(x k )|Δ xc(p)}·

Introducing the spaces (p), c(p) and c 0 (p), the author finds the first and second Köthe-Toeplitz duals of Δ (p) and asserts Δ (p) is perfect if and only if p . The necessary and sufficient conditions for an infinite matrix to transform p to c(Δ), Δ (p) to and Δ (p) to c are obtained.

Reviewer: D.Somasundaram

MSC:
46A45Sequence spaces
40C05Matrix methods in summability
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)