Some Results About Reproducing Kernel Hilbert Spaces of Certain Structure

The theory of reproducing kernel Hilbert spaces has been crucial to the development of many of the most significant modern ideas behind functional analysis. In particular, there are two classes of reproducing kernel Hilbert spaces that have seen plenty of interest: that of complete Nevanlinna-Pick spaces and de Branges-Rovnyak spaces.

In this dissertation, we prove some results involving each type of space separately as well as one result regarding their potential overlap. It turns out that a de Branges-Rovnyak space is also of complete Nevanlinna-Pick type as long as there exists a multiplier satisfying a certain identity.

Further, we extend the work of Theodor Kaluza to give sufficient conditions for having the Nevanlinna-Pick property only in terms of the coefficients of a reproducing kernel's power series. Lastly, we characterize the multiplicative shifts on any suitable de Branges-Rovnyak space by highlighting four specific properties.