On semi-homogeneous maps of degree $k$

We study properties of continuous semi-homogeneous operators of degree $k$ via various functions (e.g. measures of noncompactness) on all bounded subsets of a Banach space. We prove necessary and sufficient conditions for these functions to vanish on the image of the unit ball under these operators. In particular, we give criteriafor superposition operators to be improving and criteria for the complete continuity of the Fréchet derivatives. The results obtained can be applied in various areas of both pure and applied mathematics.