Multivariable Operator Theory: The Interplay between Function Theory, Operator Theory and Operator Algebras

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This project belongs to the branch of mathematical analysis known as operator theory. The subject was developed initially as part of the development of the mathematical foundations of quantum mechanics. Since then it has evolved in many directions, with applications not only in physics (such as the currently very active areas of quantum computing and quantum information theory), but also in electrical and mechanical engineering (where its ideas find application in e.g. the design of control systems, or in signal and image processing). A particularly new and exciting branch of this field is known as "noncommutative function theory," which has its origins in the study of certain kinds of optimization problems in engineering. It has now taken on a life of its own. The area of "multivariable operator theory" is multi-faceted but is closely connected with many questions arising in these applications (such as the study of quantum channels, and the theory of "linear matrix inequalities" in optimization). This project is aimed at expanding the array of mathematical tools available for the study of these problems. The goal of this project is to study the connection between operator theory and function theory in several variables, by analogy with the very successful one-variable theory, particularly in the rapidly developing area of "noncommutative function theory." The role of analytic function theory in the one-variable setting is very well established; e.g. a central role is played by the Cauchy transform in the unitary/circle setting or the Borel transform in the selfadjoint/real line setting. Noncommutative functions, or nc-functions for short, are meant to play a role analogous to analytic functions in the classical one-variable theory. They arise, for example, as noncommutative Cauchy transforms of "noncommutative" measures, viewed as states on a C*-algebra or related objects (such as an operator system). Problems to be studied include invariant subspaces for noncommutative operator tuples, in particular an nc-function theoretic analysis of "nc-inner" functions, and the development of a version of "noncommutative measure theory" suitable to the desired applications, in particular a version of the Lebesgue decomposition suited to the analysis, for example, of noncommutative analogs of the Cauchy transform. The project will draw on techniques from several currently active areas of mathematical analysis, with the aim of broadening and deepening understanding of the interplay between function theory, operator theory, and operator algebras. The project will also employ novel methods to study some classical objects of analysis to explore open questions in complex analysis and operator theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.