Quotients of Reproducing Kernels : Applications in Complex Analysis and Operator Theory

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 arisingfrom the well-known multiplier criterion. They are intimately connected to certain operator inequalities, such asthe famous inequality of Shimorin in sub-Bergman spaces, which extend Sarason’s sub-Hardy spaces. We developa model reminiscent of the Sz.-Nagy–Foiaş model. As an application we resolve a conjecture regarding the densityof polynomials in certain classes of weighted sub-Bergman spaces. In the second article we generalize the classicalJulia–Carathéodory theorem via reproducing kernels. We develop a new boundary notion and approach regionsto it, entirely in terms of reproducing kernels. We also introduce composition factors as a kernel-theoretic alternativeto analytic selfmaps. In the third article we identify co-isometric weighted composition operators as compositionfactors. 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 1composition factors. The set is either all analytic automorphisms of the polydisk, in which case 𝑘 is a positive powerof the Szegő kernel, or exactly the rotations composed with a permutation.