On Friday the 20th of March at 13:00 in MH:Hörmandersalen Frej Dahlin will defend his PhD thesis "Quotients of Reproducing Kernels: Applications in Complex Analysis and Operator Theory". Opponent: Professor Michael T. Jury, University of Florida.

This thesis studies reproducing kernels that are realized as pointwise quotients of two other kernels, including far-reaching generalizations of de Branges–Rovnyak spaces. In the first article, we study reproducing kernels arising from the well-known multiplier criterion. They are intimately connected to certain operator inequalities, such as the famous inequality of Shimorin in sub-Bergman spaces, which extend Sarason’s sub-Hardy spaces. We develop a model reminiscent of the Sz.-Nagy–Foiaş model. As an application we resolve a conjecture regarding the density of polynomials in certain classes of weighted sub-Bergman spaces. In the second article we generalize the classical Julia–Carathéodory theorem via reproducing kernels. We develop a new boundary notion and approach regions to it, entirely in terms of reproducing kernels. We also introduce composition factors as a kernel-theoretic alternative to analytic selfmaps. In the third article we identify co-isometric weighted composition operators as composition factors. Moreover, we extend results of Mas, Martín, and Vukotić from the unit disk to the polydisk. Specifically, under mild regularity assumptions on a reproducing kernel 𝑘 on the polydisk, we prove a dichotomy for rank 1 composition factors. The set is either all analytic automorphisms of the polydisk, in which case 𝑘 is a positive power of the Szegő kernel, or exactly the rotations composed with a permutation.