# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\phi \left(x\right)\ge 0$ when the sequence $x=\left({x}_{n}\right)$ has ${x}_{n}\ge 0$ for all n, $\phi \left(e\right)=1$ where $e:=\left(1,1,···\right)$, and $\phi \left(\left({x}_{\sigma \left(n\right)}\right)\right)=\phi \left(x\right)$ 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 $\sigma \left(n\right):=n+1$ 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 $B{V}_{\sigma }$ 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 $Ax\in B{V}_{\sigma }$ 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 $\sigma -limAx=limx$ for all $x\in bv$.
Reviewer: J.Boos

##### MSC:
 40C05 Matrix methods in summability 40C99 General summability methods 40D25 Inclusion theorems; equivalence theorems