9500870 Cowen The investigator proposes to study the structure of bounded linear operators on the Hardy and Bergman Hilbert spaces and related problems of matrix theory, approaching problems on concretely presented operators and their applications in order to gain insight into more general problems. In particular, he will continue his investigations of composition operators, slant Toeplitz operators, and related problems from matrix analysis. The project will also involve the direction of research of several undergraduate students in matrix analysis and its applications to some engineering problems with the goal of attracting young people to graduate study and research in mathematics. In this project, the investigator and his graduate students will work on several fundamental questions about composition operators on Bergman and Hardy spaces including investigation into the explanations for unexpected circular symmetry in most composition operators and seeking ways to break operators into simpler components when the symbol is a function of several variables with special structure. Further, the investigator and his graduate students will investigate the spectral theory of slant Toeplitz operators, a class of weighted composition operators with applications to smoothness criteria for wavelets. These projects will use tools from the theory of operators on Hilbert space and the theory of analytic functions in one and several variables and will build on the investigator's experience in the study of Toeplitz operators and composition operators. ***
