RUI: Complex Symmetric Operators - Theory and Applications

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The study of Hilbert space operators links a wide variety of disciplines, ranging from control theory, approximation theory, quantum mechanics, function theory, signal processing, noncommutative geometry, and matrix theory, to name but a few. The PI will study complex symmetric operators, a broad class of Hilbert space operators which, while encompassing many of the well-known and useful classes, has not been adequately studied in generality until recently. Loosely put, a Hilbert space operator is called complex symmetric if it has a symmetric matrix representation (over the complex field) with respect to some orthonormal basis. This surprisingly large class includes all normal operators, truncated Toeplitz operators (including Jordan model operators and finite Toeplitz matrices), Hankel operators, and many non-normal integral and differential operators (including the classical Volterra operator and certain auxiliary operators produced by the complex scaling method for Schrodinger operators). The PI will examine these operators at the abstract level while also considering a number of specific questions that interface with function theory, matrix analysis, and other areas. For instance, connections to complex analysis have already engaged a number of researchers from both large institutions and small colleges. The PI will collaborate with colleagues old and new, as well as sponsor undergraduate research.The emerging theory of complex symmetric operators has already proven fertile ground for undergraduate research. Many questions stemming from the proposed project are suitable for undergraduate research and the PI will recruit students from diverse backgrounds to work on them. The PI also plans to take his undergraduate researchers to conferences related to this proposal. The benefits for the students are many. For instance, they get to see mathematicians in their natural element, learn about cutting-edge research that is relevant to their own, interact with researchers in a social context, and speak candidly with graduate students ? something not available at an undergraduate institution. As part of their sustained research experience, student researchers will present posters and/or give talks in venues appropriate for undergraduate research.