Reproducing kernels in operator-related function theory

Operator-related Function Theory is a well established area of research within Analysis which combines methods from Complex Analysis and Functional Analysis, or more precisely, Operator Theory.

Complex Analysis is used to produce models for abstract (tuples of ) operators with certain properties, while, on the other hand, the powerful machinery of Functional Analysis is used to understand subtle phenomena in Complex Analysis.

The project follows the second direction and it is focused on the use of a special tool, the theory of reproducing kernel Hilbert spaces.

In recent years, a number of remarkable results have been deduced from certain algebraic properties of reproducing kernels. An important example is the  complete Nevanlinna-Pick property.

The structure of Hilbert spaces which such kernels resembles to the one of the classical Hardy space.We shall explore two important aspects in this direction: The scale of $H^p$ spaces induced by such a Hilbert space via complex interpolation and the analogue of the inner-outer factorization.

The third line of research in this project regards a much larger class of reproducing kernels, namely those which possess a complete Nevanlinna-Pick factor.

We will focus on the properties induced by such factors and also study the spaces whose kernel is obtained by dividing out this factor. These are a far-reaching generalizations of the de Branges-Rovnyak spaces, also called sub-Hardy spaces.