Multiplicative Summability Methods and the Stone-Čech Compacification

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Mathematics (HMC)

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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 ψI 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 ψI. Below (3.3), we disprove this conjecture by exhibiting a countable decreasing chain of submethods of ψI such that the intersection of their bounded summability fields is not the bounded summability field of a submethod of ψI, although it is the bounded summability field of a regular 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 ψI 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 ψI which together sum every bounded sequence of real numbers.

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