The Newlander-Nirenberg theorem states that an almost complex structure is integrable if and only if the Nijenhuis tensor vanishes. I heard that this statement is not true in infinite dimensions, since for example the Loop space of a Riemannian 3-manifold is counterexample. (In fact, I think NN fails for Fréchet manifolds in general(?)) So my question is:
Is the Newlander Nirenberg theorem valid for Banach- or Hilbertmanifolds? If not, is it possible to weaken the statement (or some conditions) such that it remains true for some class of infinite dimensional manifolds?
EDIT: Added the tag "open-problem", since NN for Hilbertmanifolds seems to be an open problem.