Matrix methods and the property of stretchings

Abstract

D. F. Dawson has proved that if x x is a sequence and A A is a matrix with convergent row sums, then there exist a stretching z z of x x and a row finite matrix B B such that A y Ay and B y By converge or diverge together for each stretching y y of z z . An extension of this result is used to answer a question proposed by D. Gaier regarding the conditions necessary for a matrix A A to have the property that if x x is a sequence with finite limit point t t , then A A sums a stretching of x x to t t .