This project will study complex symmetric operators and truncated Toeplitz operators, aiming to clarify the intriguing relationship between these two interrelated classes of Hilbert space operators. Complex symmetric operators are a broad class of operators that has not been adequately studied in generality until recently. The study of truncated Toeplitz operators, a rapidly growing branch of function-theoretic operator theory, has undergone spirited development stemming from a seminal 2007 paper of Sarason. A recent series of articles by the principal investigator and his growing list of collaborators have unearthed surprising links between these two classes. Research on this subject has the potential to be transformative, having relevance to a number of fields. For instance, connections to function theory and matrix analysis have already engaged researchers from both large institutions and small colleges. Moreover, the study of complex symmetric and truncated Toeplitz operators has already proven to be fertile ground for undergraduate research.
The study of linear operators on Hilbert space has its modern roots in the seminal work of von Neumann, who in the early twentieth century developed a rigorous framework in which to describe quantum mechanics. Since then, operator theory has found applications in electrical engineering, quantum physics, image processing, and statistics, to name a few areas outside of pure mathematics. This project will study two novel, but surprisingly ubiquitous, classes of operators, while also considering several specific questions that interface with other researchers in the mathematical sciences. To do this, the principal investigator will collaborate with colleagues old and new, as well as sponsor undergraduate research. Indeed, many questions stemming from this project are suitable for undergraduate research and the principal investigator will recruit a diverse array of students to work on them. These students will be equipped with the skills, expertise, confidence, and passion necessary to pursue careers in the mathematical sciences. This project will create not only new mathematics, but also the new mathematicians necessary to enhance and enrich the nation's infrastructure for research and education.
