Generalized averaging operators and matrix summability

Abstract:

A bounded linear operator $T$ on $C(X),X$ compact, is a g.a.o. if it has associated with it a nonnegative projection $S$ satisfying three conditions given below. An ordinary averaging operator is the case $T = S$. We show that if $T$ is g.a.o., then the following problem has a fairly neat solution: What conditions on an operator $R$ are necessary and sufficient for $\operatorname {kernel}(T) \subset \operatorname {kernel}(R)$? Application is made to the problem of the inclusion of one bounded convergence field in another, via the representation of regular matrices as linear operators on $C(\beta N/N)$.