Noninclusion theorems for summability matrices (Q1365271)

For an infinite matrix \(A=(a_{nk})\) and a sequence \(x=(x_k)\) let \(Ax\) be the sequence \((y_n)\) with \(y_n=\sum_{k=1}^\infty a_{nk} x_k\) whenever these series converge. For a sequence space \(E\) let \(E_A=\{x=(x_k)\mid Ax\text{ exists and } Ax\in E\}\). In the present paper \(E=c\) (convergent sequences) and \(E=\ell=\{x\mid\sum|x_k|<\infty\}\) are considered; \(A\) is called regular if \(c\subset c_A\) and \(\lim Ax=\lim x\) for all \(x\in c\) and \(A\) is said to be an \(\ell\)-\(\ell\) matrix if \(\ell_A\subset\ell_B\). For regular matrices \(A\) and \(B\) such that \(c_A\subset c_B\) the author proves that \(\lim_{n,k} a_{nk}=0\) implies \(\lim_{n,k} b_{nk}=0,\) where these limits are taken in the Pringsheim sense. Also, for \(\ell\)-\(\ell\) matrices \(A\) and \(B\) such that \(\ell_A \subset\ell_B\), he shows that if there exists an integer \(\mu\) and a sequence \((k(j))_{j=1,2,\ldots}\) of column indices such that \(\lim_j\sum_{n=\mu}^\infty |a_{n,k(j)}|=0,\) then \(\lim_j \sum_{n=\mu}^\infty |b_{n,k(j)}|=0\). Of course, these assertions can be phrased as noninclusion theorems, and the author does so.