Higher orderdifferentiability properties of the composition and of theinversion operator

This paper contains theorems of r-th order Frechet differentiability, with r greater-than-or-equal-to 1, for the autonomous composition operator and for the inversion operator in Schauder spaces. The optimality of the differentiability theorems for the composition is indicated by means of an 'inverse result'. A main point of this paper is that (higher order) 'sharp differentiability theorems for the composition operator can be proved by approximating the operator by composition operators whose superposing function is a polynomial, an idea which may be employed in other function space settings.