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On some new invariant matrix methods of summability. (English) Zbl 0539.40006
Let σ be a mapping of the set of positive integers into itself. A continuous linear functional ϕ on the space of real bounded sequences is a σ-mean if ϕ(x)0 when the sequence x=(x n ) has x n 0 for all n, ϕ(e)=1 where e:=(1,1,···), and ϕ((x σ(n) ))=ϕ(x) for all x . Let V σ be the space of bounded sequences all of whose σ-means are equal, and let σ-lim x be the common value of all σ-means on x. In the special case in which σ(n):=n+1 the σ-means are exactly the Banach-limits, and V σ 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 σ of sequences of σ-bounded variation, which is a Banach space. Then he characterizes all real infinite matrices A, which are absolutely σ-conservative (absolute σ-regular). Thereby A is said to be absolutely σ-conservative if and only if AxBV σ for all xbv, where bv denotes the space of sequences of bounded variation, and A is said to be absolutely σ-regular if and only if A is absolutely σ-conservative and σ-limAx=limx for all xbv.
Reviewer: J.Boos

MSC:
40C05Matrix methods in summability
40C99General summability methods
40D25Inclusion theorems; equivalence theorems