Summability of subsequences of a divergent sequence (Q2453645)

An infinite matrix \(A=[a_{ij}]\) sums a sequence \(\{x_{j}\}\) to the limit \(L\) if \(\lim_{i}\sum_{j=1}^{\infty }a_{ij}x_{j}=L\) and the matrix \(A\) is regular if it sums every convergent sequence to its limit. The author shows that any matrix \(A\) which sums all subsequences of a bounded divergent sequence cannot be regular. A property holds for almost every subsequence of a given sequence if it holds for all subsequences whose index sets have positive density (the density of a set \(S\) is defined to be \(d(S)=\lim \sup \left| \{i:i\in S,i\leq n\}\right| /n\)). The author shows that the Cesàro matrix cannot sum almost every subsequence of a bounded divergent sequence.