Mathematical Sciences: Analysis in Spaces of Analytic and Harmonic Functions

Axler 9424417 The investigators will conduct research on problems arising from the interaction between complex function theory and functional analysis. Primary emphasis will rest on the study of certain important spaces of analytic and harmonic functions and operators on these spaces. The topics to be considered include approximation by harmonic functions, iterations of the Berezin transform, hypercyclicity, Koenigs functions, composition operators, Toeplitz operators, Hankel operators and Bergman spaces. The project will also include the development of software for the symbolic manipulation of harmonic functions, appropriate for use by pure and applied mathematicians and scientists/engineers who utilize harmonic functions in their research. Operator theory has its origins in mathematical physics, in particular, in the study of quantum mechanics. The modeling of the atom required the development of non-commutative analysis. The commutator of operators is the measure of non-commutativity. Harmonic functions are encountered in the study of potential functions whose origins can be traced to electrostatics. Analytic functions are complex analogues of harmonic functions. A good portion of this project deals with operators and the analysis of their commutators that acting on spaces of analytic functions. The research on this project should be of benefit to the mathematical analysis community and to other scientists whose work involves non-commutative models and potential theory. ***

Axler 9424417 The investigators will conduct research on problems arising from the interaction between complex function theory and functional analysis. Primary emphasis will rest on the study of certain important spaces of analytic and harmonic functions and operators on these spaces. The topics to be considered include approximation by harmonic functions, iterations of the Berezin transform, hypercyclicity, Koenigs functions, composition operators, Toeplitz operators, Hankel operators and Bergman spaces. The project will also include the development of software for the symbolic manipulation of harmonic functions, appropriate for use by pure and applied mathematicians and scientists/engineers who utilize harmonic functions in their research. Operator theory has its origins in mathematical physics, in particular, in the study of quantum mechanics. The modeling of the atom required the development of non-commutative analysis. The commutator of operators is the measure of non-commutativity. Harmonic functions are encountered in the study of potential functions whose origins can be traced to electrostatics. Analytic functions are complex analogues of harmonic functions. A good portion of this project deals with operators and the analysis of their commutators that acting on spaces of analytic functions. The research on this project should be of benefit to the mathematical analysis community and to other scientists whose work involves non-commutative models and potential theory. ***