Operator Theory on the Bergman Space and the Hardy Space

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Dechao Zheng will conduct research on problems arising from the interaction between function theory and operator theory. Primary emphasis will rest on the study of Hankel operators and Toeplitz operators on the Bergman space and the Hardy space. The topics to be considered include compact perturbation of Hankel operators, function algebras and Toeplitz algebras on the disk, and reducing subspaces for multiplication operators on the Bergman space. This project focuses on the central problem of establishing relationship 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. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problems in operator theory is to classify operators satisfying additional conditions given in terms of associated operators or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.