On some new invariant matrix methods of summability. (English) Zbl 0539.40006

Let \(\sigma\) be a mapping of the set of positive integers into itself. A continuous linear functional \(\phi\) on the space \(\ell^{\infty}\) of real bounded sequences is a \(\sigma\)-mean if \(\phi(x)\geq 0\) when the sequence \(x=(x_ n)\) has \(x_ n\geq 0\) for all n, \(\phi(e)=1\) where \(e:=(1,1,...)\), and \(\phi((x_{\sigma(n)}))=\phi(x)\) for all \(x\in \ell^{\infty}\). Let \(V_{\sigma}\) be the space of bounded sequences all of whose \(\sigma\)-means are equal, and let \(\sigma\)-lim x be the common value of all \(\sigma\)-means on x. In the special case in which \(\sigma(n):=n+1\) the \(\sigma\)-means are exactly the Banach-limits, and \(V_{\sigma}\) is the space of all almost convergent sequences considered by G. G. Lorentz [Acta Math. 80, 167-190 (1948; Zbl 0031.29501)]. In a natural way the author of this paper introduces the space \(BV_{\sigma}\) of sequences of \(\sigma\)-bounded variation, which is a Banach space. Then he characterizes all real infinite matrices A, which are absolutely \(\sigma\)-conservative (absolute \(\sigma\)-regular). Thereby A is said to be absolutely \(\sigma\)-conservative if and only if \(Ax\in BV_{\sigma}\) for all \(x\in bv\), where bv denotes the space of sequences of bounded variation, and A is said to be absolutely \(\sigma\)-regular if and only if A is absolutely \(\sigma\)-conservative and \(\sigma -\lim Ax=\lim x\) for all \(x\in bv\).
Reviewer: J.Boos


40C05 Matrix methods for summability
40C99 General summability methods
40D25 Inclusion and equivalence theorems in summability theory


Zbl 0031.29501
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