Abstract
This paper contains theorems of r-th order Fréchet differentiability, with r≥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.
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