Analysis in Spaces of Analytic Functions

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Abstract Zheng Zheng 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 functions and operators on these spaces. The topics to be considered include the Hilbert transform, the Berezin transform, algebras on the unit disk, Toeplitz operators, Hankel operators, and Bergman spaces. This project focuses on a central problem in the theory of Toeplitz operators and Hankel operators, which is to establish relationships between the fundamental properties of those operators and analytic and geometric properties of their symbols. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. It 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. A good portion of this project deals with operators and the analysis of their commutators that acting on spaces of analytic functions.