  We consider Proximal Newton methods with an inexact computation of update
steps. To this end, we introduce two inexactness criteria which characterize
sufficient accuracy of these update step and with the aid of these investigate
global convergence and local acceleration of our method. The inexactness
criteria are designed to be adequate for the Hilbert space framework we find
ourselves in while traditional inexactness criteria from smooth Newton or
finite dimensional Proximal Newton methods appear to be inefficient in this
scenario. The performance of the method and its gain in effectiveness in
contrast to the exact case are showcased considering a simple model problem in
function space.
