The aim of this workshop is to explore recent advances in operator related function theory that result from the interaction with other mathematical disciplines and applied fields. The workshop will focus on major developments and open problems in the following interrelated topics.
1. Reproducing kernel Hilbert spaces and interpolation:
Recent developments reveal strong links between the algebraic structure of the kernel and the properties of the space in question. This general idea has led to important progress related to interpolation, invariant subspaces and even in multivariate operator theory, but leaves much room for deeper understanding.
2. Analysis for Dirichlet series and the Riemann zeta function:
Bohr's vision of Dirichlet series and the classical theory of Hardy spaces manifests the interplay between the additive and multiplicative structure of the integers in different guises, involving harmonic analysis, functional analysis, ergodic theory, and probability theory. Much in our understanding of distinguished operators, such as the partial sum operator, the Riesz remains to be explored.
3. Spectral theory and operator related function theory:
The spectral theory of symmetric operators (for example, Schrödinger operators) developed by Krein has found analogues in de Branges' seminal work on spaces of entire functions. A fundamental insight of Makarov and Poltoratski allows us to translate important spectral problems into questions about kernels of Toeplitz operators, or other specific objects from operator related function theory. The combination of classical tools from operator related function theory, such as Hilbert transforms, spaces of analytic functions, and techniques for Riemann-Hilbert problems, and inverse scattering theory motivates many new questions.
4. Determinental point processes and zeros of random (poly-)analytic functions:
Many determinantal point processes can be represented as the zero set of a random analytic function. The investigation of asymptotic properties, hole probabilities, and universality laws is intimately connected with the analysis of Toeplitz operators on Bargmann-Fock space. The recent advances to the poly-analytic ensembles present new challenges for operator related function and many open problems.