Majorization in de Branges spaces. III. Division by Blaschke products

This paper is a part of a series dealing with subspaces of de Branges spaces of entire functions generated by majorization on subsets of the closed upper half-plane. In the present, third, part the study of a certain Banach space generated by an admissible majorant is continued. The main theme is “invariance of the unit ball with respect to division by Blaschke products”. In connection with this topic, representability via special types of majorants plays an important role. Some (positive and negative) results on invariance under division by Blaschke factors are obtained, and the unit balls representable by $\log$-superharmonic majorants are characterized.