# Multiplicative Summability Methods and the Stone-Čech Compacification

## Document Type

Article

## Department

Mathematics (HMC)

## Publication Date

12-1959

## Abstract

A regular summability method *ψ* (for real sequences) is called multiplicative (*b-multiplicative*) if whenever *f*, *g* are two (bounded) sequences summed by *ψ*, then* ψ(fg) = ψ(f)ψ(g)*. It is known [*11*], [*17*], p. 71 that the regular multiplicative matrix summability methods are submethods of the method *ψ _{I}* corresponding to the identity matrix

*I*(i.e., methods obtained from

*I*by deleting infinitely many of its rows). It was conjectured in [

*1*] that every regular

*b*-multiplicative matrix summability method is equivalent for bounded sequences to a submethod of

*ψ*. Below (3.3), we disprove this conjecture by exhibiting a countable decreasing chain of submethods of

_{I}*ψ*such that the intersection of their bounded summability fields is not the bounded summability field of a submethod of

_{I}*ψ*, although it is the bounded summability field of a regular

_{I}*b*-multiplicative matrix summability method.

More generally, we establish a one-one correspondence between the family of regular *b*-multiplicative summability methods and the family of all closed subsets of the complement *N'* of *N* in the Stone-Čech compactification *βN *of the (discrete) space *N* of positive integers. The bounded summability field of a submethod of *ψ _{I}* corresponds to an open and closed subset of

*N'*. We are unable, however, to determine which closed subsets of

*N'*correspond to bounded summability fields of

*matrix*methods of this type, although we are able to eliminate the finite subsets.

We also construct, with the aid of the continuum hypothesis, a simultaneously consistent family of submethods of *ψ _{I}* which together sum every bounded sequence of real numbers.

## Rights Information

© 1959 Springer

## Terms of Use & License Information

## DOI

10.1007/BF01181414

## Recommended Citation

Henriksen, Melvin. 1959. Multiplicative summability methods and the Stone-Čech compacification. Mathematische Zeitschrift. 71(1):427-435.