The Associative Part of a Convergence Domain is Invariant

Abstract

Of special interest in summability theory are those conservative matrices possessing the "mean-value property". If cA={x: Ax ∊ c} denotes the convergence domain of a conservative matrix A, then A has the mean-value property in case, for each x in cA, there exists M = M(A, x) > 0 such that1This property has been considered by many writers and has been shown, among other things, to be equivalent to the requirement that the matrix be associative, i.e., for each x in cA2