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 $\mathcal{M}=G/K$. In this context, we establish the most general conditions for a bounded operator $N$ acting on $\mathrm{Lie}(G)$ to define a homogeneous vector bundle map $\mathcal{N}:T\mathcal{M}\to T\mathcal{M}$. In particular, our considerations extend all previous settings in the matter and are well suited for the case where $\mathrm{Lie}(K)$ is not complemented in $\mathrm{Lie}(G)$. We show that the vanishing of the Nijenhuis torsion for a homogeneous vector bundle map $\mathcal{N}:T\mathcal{M}\to T\mathcal{M}$ (defined by an admissible bounded operator $N$ on $\mathrm{Lie}(G)$) is equivalent to the Nijenhuis torsion of $N$ having values in $\mathrm{Lie}(K)$. As an application, we consider the question of the integrability of an almost complex structure $\mathcal{J}$ on $\mathcal{M}$ 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 $\mathrm{Lie}(G)$.