Research will be conducted on problems arising from the interaction between function theory and functional analysis. Primary emphasis will rest on the study of Toeplitz operators on some function spaces. The topics to be considered include compact operators ``closely associated with function theory'' on the Hardy space or the Bergman space, bounded Toeplitz operators, algebraic and spectral properties of dual Toeplitz operators, Hankel and Toeplitz operators on the Segal-Bargmann spaces and quasi-invariant subspaces of the Segal-Bargmann spaces. This project focuses on the central problem of establishing the relationship between the fundamental properties of those operators and analytic and geometric properties of their symbols.
The proposed work involves ideas and problems from operator theory and complex analysis. Operator Theory grew out of ideas used to study certain partial differential equations arising in physics, and became increasingly important with the advent of Quantum Mechanics. There are at least two reasons for the continuous and increasing interest in Toeplitz operators. In addition to differential operators, Toeplitz operators constitute one of the most important classes of non-selfadjoint operators and they are a fascinating example of the fruitful interplay between such topics as operator theory, function theory, harmonic analysis and operator algebras. On the other hand, Toeplitz operators are of importance in connection with a variety of problems in physics, probability theory, information and control theory, and several other fields.
