Extending Hilbert Space Operators

University Of California-San Diego, La Jolla CA

Investigators

Abstract

In classical Newtonian physics, the location and movement of a body are simultaneously measurable at a moment in time. An early discovery of modern physics was that this state of affairs is very much different at the subatomic level. In particular, the uncertainty principle of quantum mechanics asserts that it is impossible to simultaneously determine the position and momentum of a quantum particle with arbitrary precision. 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 work involved the use of operators, linear transformations acting on infinite-dimensional Hilbert spaces. Operator theory, the branch of modern mathematics that studies operators, has subsequently developed to become a far-reaching area of research in pure mathematics, and numerous additional applications throughout mathematics, physics, and engineering have been discovered. This research project involves the development of new techniques within operator theory as well as the application of established techniques to attack a number of questions involving the theory of functions. While it is not the primary focus of the project, the research has many potential applications to mathematical physics, control theory, and the theory of optimization. A pillar of modern operator theory is the Sz.-Nagy dilation theorem, which models contractions acting on Hilbert space by extending them to co-isometries acting on larger spaces. This theorem and its numerous refinements open the door to studying holomorphic functions in one and several variables through the use of operator-theoretic methods. This project studies a variety of problems in several complex variables and other areas of analysis using these operator-theoretic methods. In particular, operator-theoretic methods will be employed to study: interpolation problems of Nevanlinna-Pick and Cartheodory-Fejer type; the boundary behavior of analytic functions defined on polydiscs and poly-halfplanes; the derivation and descriptive theory of extremal holomorphic mappings arising from the Caratheodory and Kobayashi extremal problems; and the canonical derivation of representation formulae for analytic functions in specific classes such as the Schur, Herglotz, Pick, Loewner, Bessmertnyi, and Stieltjes classes on domains in several variables. A related focus of the research is to apply modifications of the modeling methods to develop the theory of analytic functions on varieties in commuting variables and on free domains in several non-commuting variables.