In this report we come back to the method of factorization of a second order elliptic boundary value problem presented in . In this paper, it was shown that, in the case of a cylinder, the boundary value problem can be factorized in two uncoupled first order initial value problems. This factorization utilizes the Dirichlet to Neumann operator which satisfies a Riccati equation. Here we consider Hilbert-Schmidt operators, a framework already used by R. Temam which provides tools for a direct study of this Riccati equation.