The conservative matrix on locally convex spaces. (Q1432842)

Let \((X,\tau_X)\) and \((Y,\tau_Y)\) be locally convex spaces, \(c(X)\) and \(c(Y)\) the spaces of all \(X\)-valued and \(Y\)-valued convergent sequences, respectively, and \(A=(A_{ij})\) an operator-valued infinite matrix with \(A_{ij}\in L(X,Y)\). In the case of Fréchet spaces \(X\) and \(Y\), \textit{M. S. Ramanujan} [Math. Ann. 159, 365--373 (1965; Zbl 0139.08303)] characterized those matrices \(A\) which are conservative, that is, which transform \(c(X)\) into \(c(Y)\,.\) Moreover, \textit{L. W. Baric} [Stud. Math. 39, 165--180 (1971; Zbl 0212.45602)] considered in this context the so-called \(\chi\)-function on the set of all conservative matrices \(A\) and proved that a conservative matrix \(A\) is conull if and only if \(\chi(A)=0.\) The main objective of this paper is to extend the mentioned results to the case of barrelled spaces \(X\) and \(Y\) having some additional properties.