  We develop a globalized Proximal Newton method for composite and possibly
non-convex minimization problems in Hilbert spaces. Additionally, we impose
less restrictive assumptions on the composite objective functional considering
differentiability and convexity than in existing theory. As far as
differentiability of the smooth part of the objective function is concerned, we
introduce the notion of second order semi-smoothness and discuss why it
constitutes an adequate framework for our Proximal Newton method. However, both
global convergence as well as local acceleration still pertain to hold in our
scenario. Eventually, the convergence properties of our algorithm are displayed
by solving a toy model problem in function space.
