A Sufficient Condition for a Regular Matrix to Sum a Bounded Divergent Sequence

If a matrix .4 transforms 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.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 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