Mathematical Sciences: Operator Theory and Analysis
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Operator theory is a major discipline in Modern Analysis.               Its origins lie in the study of mathematical physics and partial                differential equations in the early twentieth century.                          Subsequently, the subject of operator theory has grown to a                     central position in such investigations, and in core mathematics                as well.  Indeed, since many explicit operators arise in complex                function theory, differential geometry, and approximation theory,               there is a broad range of applicability.  In particular, the                    connection between operator theory and complex function theory                  has long been actively pursued from various points of view.  The                recent brilliant innovations by deBranges that led to the                       solution of the Bieberbach conjecture have further developed the                deep interdependence of these disciplines.  Professors Rosenblum                and Rovnyak form a research team that has had an excellent record               of research on this interface, and their earlier work was in a                  distant way a precursor of deBranges' ideas.  Both singly and                   together, their successes include seminal results in perturbation               theory of unitary transformations, spectral theory of the Hilbert               matrix, factorization and interpolation theory for bounded                      analytic functions, a generalization of the corona theorem, and                 extensions of the basic inequality in deBranges' work (loc cit).                        The current proposal seeks to extend the estimation theory              of Riemann mapping functions that produced the proof of the                     Bieberbach conjecture.  Known extensions treat powers greater                   than -3/2, and it is proposed to explore nonpositive integral                   powers.  Other extensions involve interpolation theory, and have                applications in operator theory.  Professors Rosenblum and Rovnyk               will also study interpolation theory for Newton spaces, which are               finite difference analogs of Hardy spaces.  Emphasis will be                    placed on the special functions related to Newton spaces, and to                logarithmic versions of these spaces.  The project also includes                operator equations associated with analytic functions of bounded                mean oscillation and with the corona theorem.
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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