Differentiability with respect to parameters in global smooth linearization

Let X be a Banach space and Tθ:X→X a family of invertible contractions, Tθ=Lθ+fθ, where Lθ is linear and fθ is nonlinear with fθ(0)=0. We give conditions for the existence of a family of global linearization maps Hθ, such that Hθ∘Tθ∘Hθ−1=Lθ, with a smooth dependence on θ. The results depend strongly on the choice of some appropriate spaces of maps, adapted norms and the use of a specific fixed point theorem with smooth dependence on parameters.