Collaborative Research in Operator Theory on Holomorphic Function Spaces

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Professor Shapiro will continue his long-term collaboration with Paul Bourdon on problems arising from the interaction between the modern theory of linear operators and the classical theory of analytic functions. Shapiro and Bourdon will study problems involving norms, numerical ranges, decomposability, and chaotic behavior for operators that occur naturally on spaces of analytic functions, most notably composition operators and operators that commute with backward shifts.The theory of linear operators originates from mathematical physics, especially from quantum mechanics, and has more recently found significant application in control system theory. The numerical range is a convex subset of the plane that has proven useful to engineers in determining the stability of control systems associated with linear operators. The chaotic behaviour of dynamical systems has recently become an important subject spanning mathematics, physics, engineering, and biology. It has only recently been discovered that significant classes of linear operators can give rise to chaotic systems, and this has led to unexpected connections between operator theory and dynamical systems. The idea of decomposability---the study of how to break a complicated linear system up into simpler ones---has recently been shown to have surprising connections with such chaotic behaviour. The research of Shapiro and Bourdon arises from, and has contributed to, these developments: it is anticipated that the project outlined in this proposal will further enhance our understanding of the connection between dynamical systems and operator theory.