Continuous dependence of solutions of operator equations. I

Abstract

Continuous dependence of the solutions of the operator equation x = T x + z x = Tx + z in a topological vector space is the main subject of the paper. We find sufficient and necessary conditions for the continuous dependence on the data (T, z) or on a parameter. We do it for the space of all closed operators. Equivalent conditions for particular subfamilies are discussed. Among other families we deal with compact operators, compact perturbations of the identity, condensing operators and demicompact operators.