Composition operators and Banach spaces – Comop

This project aims to study composition operators in two different frameworks. The first one is that of Lipschitz free spaces: given a Lipschitz map between two pointed metric spaces, we have to compare the properties of this map with that of the associated linear operator between the Lipschitz free spaces over these metric spaces. We are also interested in holomorphic Lipschitz free spaces.
Our second framework is that of holomorphic function spaces. On the one hand, we are interested in de Branges-Rovnyak spaces H(b). The study of composition operators over these spaces was recently initiated, when b is a rational function. We plan to do a systematic study, without this restriction on b, studying for instance the boundedness, the compactness, the spectrum or the dynamical properties of the composition operator.
On the other hand, we are interested in spaces of holomorphic functions in several variables, and even in an infinite number of variables through the spaces of Dirichlet series. The study of composition operators is much more difficult in that case, and many problems have to be better understood.