Author’s abstract: We study the well-posedness in the space of continuous functions of the Dirichlet boundary value problem for a homogeneous linear second-order differential equation \(u'' + Au = 0\), where \(A\) is a linear closed densely defined operator in a Banach space. We give necessary conditions for the well- posedness, in terms of the resolvent operator of \(A\). In particular we obtain an estimate on the norm of the resolvent at the points \(k^2\), where k is a positive integer, and we show that this estimate is the best possible one, but it is not sufficient for the well-posedness of the problem. Moreover we characterize the bounded operators for which the problem is well-posed.