Mathematical Sciences: Structure of Toeplitz and CompositionOperators

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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. A particularly important class of such operators are referred to as "subnormal". They are significant for two reasons. First, there is enough residual normal information to attempt a deeper understanding of their inherent structure. Recent results, such as the fact that subnormal operators are reflexive and have invariant subspaces, have given an added impetus to the study of their spectral properties. Secondly, since many explicit operators that arise in complex function theory, differential geometry, and approximation theory are subnormal, there is a broad range of applicability. Professor Cowen is a leader in the spectral theory of subnormal operators and its relationship to classical analysis. His program involves the investigation of explicit operators on the Hardy space of the unit disk in order to gain insight into more general problems. In particular, he proposes to continue his investigation of composition operators, Toeplitz operators, and related operators. In recent collaborative work he found a class of composition operators which are subnormal. Professor Cowen proposes to extend this class, and to further investigate subnormality. Operators more general than subnormal arise as multiplication operators on Krein space. Professor Cowen proposes to initiate the study of such operators. In earlier work, he solved an old problem by constructing subnormal Toeplitz operators which are neither normal nor analytic. Professor Cowen proposes to investigate the more prevalent hyponormal Toeplitz operators with the goal of finding conditions on the symbol that determine hyponormality.