Operator Related Function Theory

Reproducing kernel Hilbert spaces have been studied for decades, but remain an expanding and vivid area of research. Recent developments reveal strong links between the algebraic structure of the kernel and the properties of the space in question. This general idea is certainly not completely understood, but it has led to important progress related to interpolation, invariant subspaces, and also in multivariate operator theory, which combines complex analysis in several variables and the study of commuting tuples of operators.

A more special related topic of recent interest are the Hardy spaces of Dirichlet series. It began with a paper of Hedenmalm, Lindqvist, and Seip in 1997. An interesting theory is now evolving, modeled on the classical theory of Hardy spaces but also interacting profoundly with several complex variables and analytic number theory. One of the most difficult research objectives in the development of this theory, is to achieve a deeper understanding of the mapping properties of Riesz projections and similar operators on the infinite-dimensional torus.

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. It has been recognized that de Branges' structure theorems for these spaces have valuable applications in this field, and in recent years, Makarov and Poltoratski have introduced more general model spaces to translate important spectral problems into questions about kernels of Toeplitz operators, or other specific objects from operator related function theory. Their work opened a wide circle of interesting new problems. In addition, scattering theory combines methods of mathematical physics, operator related function theory, partial differential equations, and harmonic analysis. This field originated in the study of basic physical phenomena and finds direct applications in modern technology, with uninvasive control and nanotechnology taken as examples. Classical tools from operator related function theory, such as Hilbert transforms, spaces of analytic functions, and techniques for Riemann-Hilbert problems are widely used, and in many cases the inverse scattering problem serves to motivate new questions.

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At a glance
March 26, 2012 — March 31, 2012