Summability in Amenable Semigroups

Abstract

Abstract. A theory of summability is developed in amenable semigroups. We give necessary and (or) sufficient conditions for matrices to be almost regular, almost Schur, strongly regular, and almost strongly regular. In particular, when the amenable semigroup is the additive positive integers, our theorems yield those results of J. P. King, P. Schaefer and G. G. Lorentz for some of the matrices mentioned above.

Matrix Summability in Amenable Semigroups

“…That such a sequence always exists follows from Theorem 3 in [3] and Lemma 5.1 in [6]. Now by Proposition 4.4 in [7], \s~asfe C\(K) for every/e m(S). Hence, using Theorem 8 in …”

“…That the conditions are sufficient was proved in Theorem 7.3 in [7]. To show that they are necessary we shall only check (4.3.3) since the other two are easy.…”

“…We shall freely use notations and definitions in [7]. Recall that a semigroup S is left amenable (LA) if the Banach space of all bounded real-valued functions on 5 with the sup norm, m(S), has a normalized positive left translation invariant linear functional.…”

“…Introduction. In [7] the author discusses various methods of matrix summability in amenable semigroups. In that paper, sufficient conditions were given for an infinite matrix to be almost Schur and almost strongly regular.…”

Asymptotically Statistical Equivalent of Orderαin Amenable Semigroups

“…Any sequence of finite subsets of G satisfying (i), (ii) and (iii) is called a Folner sequence for G. For a detailed account of amenable semigroups one may refer to ( [5], [6], [7], [17], [18])…”