Nijenhuis operators on Banach fibration

In the infinite-dimensional Banach setting, we consider general smooth Banach fibrations $\tau:M\to M_0$ and `$(1,1)$-tensors' $N:TM\to TM$ that are projectable (in the obvious sense) onto Nijenhuis operators $N_0:TM_0\to TM_0$ on $M_0$. We prove that the vanishing of the Nijenhuis torsion of $N_0$ is equivalent to the fact that the Nijenhuis torsion of $N$ takes only vertical values, i.e., values in $ker(T\tau)$. Consequences for almost complex structures on (real) Banach manifolds are also derived. As canonical examples, we define tangent lifts $d_T(N_0):TT M_0\to TT M_0$ of Nijenhuis operators $N_0$ in the Banach category, and prove that they are automatically projectable for the canonical fibrations $\tau_{M_0}:TM_0\to M_0$. Finally, we comment on the projectability in the case of Banach homogeneous manifolds $\tau:G\to G/K$, studied recently by some authors.