A Sufficient Condition for a Regular Matrix to Sum a Bounded Divergent Sequence
A. Mary Tropper · Proceedings of the American Mathematical Society · 1953
If a matrix .4transforms a sequence {z"} into the sequence { z, A is said to be regular.The well known necessary and sufficient conditions for A to be regular are1 (a) X^-i |g»,*| Wo, (b) limn^oo an,k = 0 for every fixed fe, (c) XXi a»j»«A»-»1 as w-»°o.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.3It 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.4R. P. Agnew has given a simple sufficient condition that a regular matrix shall sum a bounded divergent sequence.He has proved5 that if A is a regular matrix such that lim",t,M On,t = 0, then some divergent sequences of O's and 1's are summable-^4.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 Cesàro matrix (C, 1) by the corresponding row of the unit matrix, given by