Richter and Sundberg will continue their research in Function-Theoretic Operator Theory and Model Theory based on Agler's idea of extremals in families of operators. One particular emphasis in their work will be a study of Hilbert modules based on the Drury-Arveson space of analytic functions in several complex variables.
The study of spaces of analytic functions has a long and rich history as a meeting ground and source of ideas from a wide range of areas in both Pure and Applied Mathematics such as Complex Analysis, Harmonic Analysis, Operator Theory, Functional Analysis, Control theory, and Partial Differential Equations. Operator theory has its roots in the work on Partial Differential Equations of Fredholm and Hilbert in the late nineteenth and early twentieth centuries. The study of spaces of analytic functions has its origins in the work on the classical Hardy spaces by Hardy, Fischer, and the Riesz brothers, among others, in the first half of the twentieth century. The two areas met in the 1940's in the work of A. Beurling on the unilateral shift, which yielded a complete structure theory of an important infinite dimensional operator using Hardy-space techniques. The generalization of Beurling's results to arbitrary multiplicity shifts together with the Sz.Nagy dilation theorem is the basis for a model theory for contraction operators on Hilbert spaces. Thus, up to scaling, every bounded linear operator on a separable Hilbert space can be modeled using a semi-invariant subspace of a vector-valued Hardy space. As many naturally occurring processes can be modeled by use of such linear operators, this has applications that can be felt throughout science. More research is needed and is being done to clarify the model theory. In particular, there has been a large effort devoted to extending the ideas developed in the study of the Hardy spaces to other spaces of analytic functions, and in particular to the multivariable setting. The current work of Richter and Sundberg is in this area.
The research conducted by the P.I.’s on the grant involve the interrelated fields of Complex Analysis and Operator Theory, as well as applications of ideas and tools in these fields and other areas, such as Harmonic Analysis and Mathematical Physics. Complex Analysis, the study of complex analytic functions is a classical area with origins tracing back at least to the early 19th century. This was a time when Mathematics and Physics were not as separate disciplines to nearly the extend that they are today, and Complex Analysis from its inception to the present time has had important applications in Physics and Engineering. Operator Theory is of more recent vintage with roots in the latter part of the 19th century. It grew considerably in scope and importance in the early 20th century with the rise of abstraction in Pure Mathematics and even more with the advent of Quantum Mechanics. In Quantum Mechanics one models a physical system by assigning to it a "state space" which is an abstract infinite-dimensional complex vector space (a "Hilbert space"). Two striking differences from Classical Physics appear. Firstly, the appearance of complex numbers. Secondly in Quantum Mechanics measurements are modelled by the application of certain operations, called "Self-Adjoint Operators", on the state space. These operations are noncommutative – it matters the order in which they are performed, which corresponds to the physical fact observed in microscopic systems that a measurement inevitably disturbs a system, so that the order of performance of two different measurements can make a difference. The mathematical field of Operator theory studies these and related operators. It continues to have a rich interaction with Quantum Mechanics, but it also has many applications in other areas of Physics and Engineering, as well as applications in various areas of Pure Mathematics. Intellectual Merit: The research conducted by the P.I.’s on this grant mainly concerns interactions between Complex Analysis and Operator Theory. Specifically they studied operators modelled by natural operations in Complex Analysis and aimed to use properties of analytic functions to gain as complete a description as possible of these operators. The prototype for this work is Beurling’s Theorem, which provides a complete description of an important operator called the Unilateral Shift, and which is modelled by the operations of multiplication by the coordinate function z on the Hardy space H2 of functions analytic in the unit disc in the complex plane. The research carried out on this grant includes projects aimed at extending Beurling’s Theorem to other spaces of analytic functions, both in one and several complex variables. For example questions concerning the structure of the commutative d-tuple of operators of multiplication by the coordinate functions z1, z2, …, zd on various spaces of functions analytic in the unit ball of Cd were studied, and a structure theorem for certain 2-isometric operator tuples was discovered. Other results obtained under the grant include: A study of the so-called "space of weak products" of Dirichlet functions. Spaces of weak products arise naturally in the context of a class of operators called Hankel operators, another kind of operators modelled using spaces of analytic functions. This work includes the first systematic study of abstract weak product spaces. Research of the "Transitive Algebra Conjecture", which is a strengthened version of the Invariant Subspace Conjecture. A strong converse to a result due to del Rio, Makarov, and Simon, and independently to Gordon, concerning rank-one perturbations of self-adjoint operators. This work is quite directly involved with Quantum Mechanics and was a geometric construction in Complex Analysis. Broader Impact: The P.I.’s used this project as an opportunity to train and support the Ph.D. work of several graduate students. Furthermore, the P.I.’s gave many presentations about their work to audiences comprised of professionals working on related problems in mathematics.
