Zerlegungen von Wachstumsbereichen und Wirkfeldern für die Verfahren bewichteter Mittel. (Decompositions of growth domains and effect fields for the method of weighted means) (Q1089538)

Let A be a permanent matrix summability method and suppose there exists a sequence \((\omega _ n)\) such that \(x_ n=o(\omega _ n)\) is satisfied whenever x is A-summable. We ask whether it is true that, under these conditions, every sequence x satisfying \(x_ n=o(\omega _ n)\) admits a decomposition \(x=y+z\), where y and \(((-1)^ nz_ n)\) are both A- summable. We give a positive answer for a class of permanent summability methods A including the methods \(M_ p\) of weighted means. Let A be a permanent matrix summability method and suppose there exists a sequence \((\omega _ n)\) such that every A-limitable sequence x necessarily satisfies \(x_ n=o(\omega _ n)\). Is it true that every A- limitable sequence x admits a decomposition \(x=y+z\), where y is convergent and \((z_ n/\omega _ n)\) is summable? We prove that again the answer is in the positive for the above class of summability methods. In both cases, the decomposition is established using functional analytic methods. In the first case, we make use of the fact that \(c_ 0\) is a BK-space with the Wilansky property [see \textit{G. Bennett}, Math. Z. 194, 321-330 (1987) or the second author, Arch. Math. 48, 149-152 (1987)]. In the second case, the decomposition relies on the fact that \(c_ A\) has the Wilansky property for the class of summability methods A involved. The latter fact is a consequence of a significant generalization of the Bennett/Stadler result, obtained by the authors and destined to appear elsewhere.