Mathematical Sciences: The Nevanlinna Counting Function, Linear-Fractional Models, and Orbits of Operators

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ABSTRACT Bourdon The project entails an investigation of function-theoretic issues associated with the following question posed by Walter Rudin (in 1988): when will the set of nonnegative powers of a bounded, holomorphic function on the open unit disk constitute an orthogonal subset of the classical Hardy-two space? Previous work by the project's principal investigator suggests that Rudin's question may be answered through an analysis of properties of the Nevanlinna counting function and radial subharmonic functions. The project also involves a continuation of the principal investigator's study of the behavior of functions and operators under iteration: chief objectives here are to characterize the cyclic behavior of a certain class of composition operators acting on the classical Hardy space and to resolve the invariant-subspace problem for hyponormal operators on Hilbert space. Methods to be employed include computer studies of the Nevanlinna counting function (carried out by undergraduates) and an analysis of the extent to which information about orbits of a linear operator can contribute to the establishment of the existence of a nontrivial, invariant subspace for that operator. The project focuses on problems in the areas of function theory and operator theory. The goal of work in function theory is to provide information and tools that enable scientists to understand/predict how functions of certain types will behave in various situations. For example, the study of the behavior of functions under iteration (which forms a component of the project) connects function theory to the analysis of dynamical systems, including systems that behave chaotically. The goal of much current research in operator theory (including the operator-theoretic component of the project) is to determine when a given linear operator preserves a small part of the whole space on which it acts. When smaller parts are preserved, one hopes the whole operator may be understood in terms of its simpler parts. Linear operations include simple processes such as rotation in space and complex processes such as the application of Schroedinger's equation of quantum mechanics to certain function spaces. The project will also contribute to the development of the nation's human-resources: three undergraduate students will receive significant training and research experiences through their participation in the project.