On the matrix transformations of absolute summability fields of reversible matrices (Q1320485)

Let \(A\) be a given reversible matrix (namely, \(z=Ax\) has a unique solution \(x\) for each \(z\in c\)); let \(e:= (1,1,\dots)\), \(e^ n:= (e_ r^ n):= (0,\dots, 0,1,0, \dots)\), and let the sequence \(\eta\) and the matrix \(H:= (\eta_{kr})\) be defined by \(e= A\eta\) and \(e_ r^ n= (AH)_{nr}\). Let \(B\) be a given triangular matrix. It is the first purpose of this paper to find necessary and sufficient conditions on a general matrix \(M\) in order that \(M\) should map any absolutely \(A\)- limitable sequence \((x\in bv_ A)\) into one whose \(M\)-transform is \(B\)- limitable \((Mx\in c_ B)\). Writing \(\Gamma:= (\gamma_{nk}):= (BM)H\), it is shown (Theorem 1) that \(bv_ A \subseteq (c_ B)_ M\) if and only if the relevant sums exist and (i) \(\eta\in c_{BM}\), (ii) \(\exists \lim_ n \gamma_{nk}\) for each \(k\), (iii) \(\sum_{k=0}^ r \gamma_{nk}= O(1)\) for every \(n\) and \(r\). A second theorem examines conditions for \(bv_ A \subseteq (bv_ B)_ M\) and the results are applied to the case where \(A\) is a Riesz (weighted mean) matrix.