Mathematical Sciences: Topics in Operator Theory

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Operator theory has its origins in the mathematical foundations of quantum mechanics. There, one needs infinitely many degrees of freedom (best realized in infinite-dimensional Hilbert space) and a way of representing physical quantities by objects that do not obey the ordinary laws of arithmetic. Operators on Hilbert space serve the latter purpose quite well. A given operator moves around pieces (subspaces) of the underlying space. One of the fundamental questions in the subject is, how much can you tell about the operator just from its invariant subspace lattice (the collection of subspaces that are mapped into themselves by the operator)? Important discoveries by the principal investigator on this project have shown that the relationship is more subtle than was hitherto conjectured. This work will be followed up. Another aspect of the research to be performed has to do with a particular way of realizing Hilbert space and certain operators on it. Namely, the space comes as a space of analytic functions of several complex variables, all of them defined on some common domain, and the operator comes from composing the functions with a transformation of the domain. When more than one complex variable is involved, well-behaved transformations of the domain can give rise to composition operators that are not well behaved. Exploration of this mysterious phenomenon is part of the agenda of the project.