  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 Fr\'echet derivatives. The results
obtained can be applied in various areas of both pure and applied mathematics.
