Consider the operator equation, AX − XB = Q(∗), in which A, B, Q are appropriately given bounded or unbounded linear operators in a Hilbert space H . Let V A , V B be domains of A, B, respectively. The equation (∗) has a unique solution X = X Q in L ( V B , H ) if: (i) a countable orthonormal basis { b i| i ϵ N } exists for V B such that each b i is an eigenvector of B belonging to an eigenvalue μ i ; (ii) a sequence { V A(n(k))|k ϵ N } of finite-dimensional subspaces of V A exists such that V A(n(k)) contains A[ V A(n(k))] and the space spanned by {Q b i|1 ⩽ i ⩽ n(k)} , for all; (iii) for every i, the operator T i ≡ A − μ i I in V has a continuous inverse defined on its image and (iv) {T i} iϵ N is an l 2-sequence or an appropriately weighted l p -sequence. A numerical analytical approach to this theorem is outlined using internal approximations and reflexivity of certain spaces. It is discussed why the existence of solution to a properly defined dual problem to (∗) implies a certain one-sided coercivity condition which, in turn, leads to the existence of a unique solution to equation (∗). Examples are given.