The gamma function which is expressed by an improper integral is used to establish the fractional difference operators and fractional Banach sets. In this study, we achieve some comprehensive and complementary results related to characterizations of the matrix classes of fractional Banach sets. We also obtain some identities or inequalities for the Hausdorff measure of noncompactness of the corresponding matrix operators, and finally find the necessary and sufficient conditions for those matrix operators to be compact.