Extending Hilbert Space Operators

University Of California-San Diego, La Jolla CA

Investigators

Abstract

In classical Newtonian physics, the position and momentum of a body are assumed to be simultaneously knowable at an instant in time. An early discovery of modern physics was that this situation is very much different at the subatomic level. For example, the famous Uncertainty Principle of Werner Heisenberg asserts that it is impossible to measure simultaneously the position and momentum of a quantum particle such as an electron orbiting the nucleus of an atom. In 1926, von Neumann laid the precise mathematical foundation for what it is that can actually be measured in the case of a subatomic particle. This seminal breakthrough involved the use of "operators," linear transformations acting on infinite dimensional Hilbert spaces. Operator theory, the branch of modern mathematics that studies operators, has grown over the last eighty-eight years to become a far-reaching area of research in mathematics that has had a major impact on many areas of mathematics, physics, and engineering. This project involves the development of new techniques within operator theory, as well as the application of established techniques, to attack a number of problems within mathematics. Though it is not the major focus of the project, the research has many possible applications to both mathematical physics and control theory. A pillar of modern operator theory is the Sz.-Nagy Dilation Theorem, which models a contraction acting on Hilbert space by extending it to a co-isometry acting on a larger space. This theorem and its numerous refinements open the door to studying analytic functions in one and several variables through the use of operator-theoretic methods. The principal investigator will study a variety of problems in several complex variables and other areas of analysis using these operator-theoretic methods. In particular, he will use operator-theoretic methods to study the following: interpolation problems of Nevanlinna-Pick and Cartheodory-Fejer type; the boundary behavior of analytic functions defined on polydiscs and polyhalfplanes; the derivation and descriptive theory of extremal holomorphic mappings arising from the Caratheodory and Kobayashi extremal problems; and the canonical derivation of representation formulas for analytic functions in specific classes such as the Shur, Herglotz, Pick, Loewner, Bessmertnii, and Stieljes classes. A related focus of the research is to apply the commutative modeling methods to develop the theory of analytic functions in several noncommuting variables.