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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 $\ell\sp{\infty}$ of real bounded sequences is a $\sigma$-mean if $\phi(x)\ge 0$ when the sequence $x=(x\sb n)$ has $x\sb n\ge 0$ for all n, $\phi(e)=1$ where $e:=(1,1,...)$, and $\phi((x\sb{\sigma(n)}))=\phi(x)$ for all $x\in \ell\sp{\infty}$. Let $V\sb{\sigma}$ 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 $V\sb{\sigma}$ is the space of all almost convergent sequences considered by {\it G. G. Lorentz} [Acta Math. 80, 167-190 (1948; Zbl 0031.29501)]. In a natural way the author of this paper introduces the space $BV\sb{\sigma}$ 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 $Ax\in BV\sb{\sigma}$ 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

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