A sufficient condition for a regular matrix to sum a bounded divergent sequence

Abstract

If a matrix A transforms a sequence { zn } into the sequence { n} i.e., if On= D-1 an,kZkg and if n->z as n-> oo whenever zn--z, A is said to be regular. The well known necessary and sufficient conditions for A to be regular are' (a) Et' | an,k| no, (b) limn,o an,k = O for every fixed k, (c) Z. 1 an,k-An-*l as n-* oo. It is known2 that if a regular matrix sums a bounded divergent sequence, then it also sums some unbounded sequence. The converse is, however, false.3 It is consequently of interest to find sufficient conditions for a regular matrix to sum a bounded divergent sequence. Many authors have considered summability of bounded sequences.4 R. P. Agnew has given a simple sufficient condition that a regular matrix shall sum a bounded divergent sequence. He has proved that if A is a regular matrix such that limn,ko an,k =0, then some divergent sequences of 0's and l's are summable-A. There are, however, very many simple regular matrices which do not satisfy this condition, but which are known to sum a bounded divergent sequence. For example, the matrix A obtained by replacing every third row of the Cesaro matrix (C, 1) by the corresponding row of the unit matrix, given by