THE CONSERVATIVE MATRIX ON LOCALLY CONVEX SPACES

Abstract

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be locally convex spaces, $c(X)$ and $c(Y)$ the $X$-valued and $Y$-valued convergent sequence spaces, respectively, $A_{ij} \in L(X,Y)$ and $A=(A_{ij})$ an operator-valued infinite matrix. In this paper, we characterize the matrix $A=(A_{ij})$ which transforms $c(X)$ into $c(Y)$. As its applications, we introduce the chi function $\chi$ on locally convex spaces, and show that a conservative matrix is conull if and only if $\chi$ (A)=0$.