Nijenhuis operators on Banach homogeneous spaces

Abstract

For a Banach–Lie group G and an embedded Lie subgroup K, we consider the homogeneous Banach manifold M D G=K. In this context, we establish the most general conditions for a bounded operator N acting on Lie.G/ to define a homogeneous vector bundle map N W T M ! T M. In particular, our considerations extend all previous settings in the matter and are well suited for the case where Lie.K/ is not complemented in Lie.G/. We show that the vanishing of the Nijenhuis torsion for a homogeneous vector bundle map N W T M ! T M (defined by an admissible bounded operator N on Lie.G/) is equivalent to the Nijenhuis torsion of N having values in Lie.K/. As an application, we consider the question of the integrability of an almost complex structure J on M induced by an admissible bounded operator J , and we give a simple characterization of the integrability in terms of certain subspaces of the complexification of Lie.G/.