The P.I. will study complex symmetric operators, a class of Hilbert space operators that, 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, compressed 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).
Recent results of the P.I. and collaborators include a general structure theorem for complex symmetric operators and new variational principles for singular values of compact complex symmetric operators that indicate the fundamental role played by certain antilinear symmetries. The proposed project aims to continue the general study of complex symmetric operators (or of certain subclasses thereof) while also focusing on the applications of existing results to related areas (for instance function theory and matrix theory). To broaden the impact and applicability of this work, the P.I. will collaborate with his colleagues in mathematics, physics, and engineering as well as sponsor undergraduate research related to the project. It is hoped that these combined efforts and ensuing results will be significant for current studies in operator theory, function theory, matrix theory, and some specific branches of mathematical physics and engineering.
