Mathematical Sciences: Subnormal Operators and Related Topics

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Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. Thus, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum theory, etc. could be studied productively via the integral equations that model the phenomena. So it has been, that from the fertile minds of Hilbert, von Neumann, and other giants that the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. At the heart of this methodology is the deep investigation of the spectrum of an operator and the concommitant study of its invariant subspaces. For the so-called self-adjoint operators, this theory is now a standard technique throughout analysis, and the spectral theorem provides the necessary building blocks for all such operators. Although not quite as complete, spectral theory is significantly well understood for an extensive generalization of the self-adjoint case; viz., the "normal" operators. The current frontier, therefore, in the study of this structure theory rests in the non-normal theory. Professors Conway has long been a leader in the study of subnormal operators, a broad class of operators that arise in applications, and for which there is enough residual normal information to attempt a deeper understanding of their inherent structure. Recent results, suchas the fact that subnormal operators are reflexive and have invariant subspaces, have given an added impetus to the study of their spectral properties. Since many explicit operators that arise in complex function theory, differential geometry, and approximation theory are subnormal, there is a broad range of developing interaction. Professor Conway will study subnormal operators that arise naturally in actions on Hardy spaces, thereby exploring the interaction with function theory. He will also investigate the internal structure theory of subnormal operators, with particular consideration of unitary equivalence, bundle shifts, and the relationship with general operator theory.