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.
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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