zbMATH — the first resource for mathematics

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

MSC:
 40C05 Matrix methods for summability 40C99 General summability methods 40D25 Inclusion and equivalence theorems in summability theory
Zbl 0031.29501
Full Text: