Reproducing kernel Hilbert spaces and dilation theory

• Applicant: Professor Dr. Michael Hartz
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 466012782

Operator theory and complex analysis are two branches of mathematical analysis that have experienced fruitful interactions for a long time, with each subject providing key insights into the other. The proposal addresses questions at the interface of these two areas, by additionally drawing from operator algebras and harmonic analysis. In particular, the proposal concerns problems about reproducing kernel Hilbert spaces and dilation theory. In the area of reproducing kernel Hilbert spaces, the main focus lies on Hilbert spaces of holomorphic functions, especially complete Pick spaces. The basic objective in dilation theory is to represent an operator on a Hilbert space as a piece of a better understood operator on a larger Hilbert space. The research programme addresses several concrete problems of relevance for both reproducing kernel Hilbert spaces and dilation theory.The best understood complete Pick space is the Hardy space; the long term vision is to bring the theory of complete Pick spaces to a similar level of sophistication. A central goal of the proposal is to contribute to this long term vision. This is expected to yield insights into classical spaces such as the Dirichlet space or the Drury--Arveson space, also known as bosonic Fock space. Using tools from a number of areas that recently became available, such as non-commutative function theory, operator space techniques and realization formulas, the project group will tackle several problems, some of which proved resistant to classical approaches. This includes interpolating sequences, cyclicity questions and the validity of von Neumann's inequality for commuting contractions up to a constant.

• DFG Programme: Independent Junior Research Groups