The core of this proposal is to systematically study certain interactions between holomorphic function spaces and operator theory. These holomorphic spaces include Dirichlet spaces, potential capacity related scale space which covers Bloch space and BMOA, and the analytic Morrey spaces. The operators are multipliers, bilinear forms, such as Hankel operators and Toeplitz operators, and commutators. The study involves the following objects: the orthogonal projection, the point evaluation operator, mean value operator, the differential operator such as d-bar, Laplacian and Dirac operators, Clifford analysis, atomic decomposition, approximation numbers, predual, intermediate spaces, Carleson measures and potential capacity. Techniques in harmonic analysis, complex analysis and operator theory will bebe used and developed in this study.

Beside its own charm in pure mathematical research, the interaction between holomorphic function spaces and operator theory has many applications in several fields of applied sciences. Automatic control, navigation and system design in engineering are examples of the applications. Operator theory has its roots in the rigorization of quantum mechanics and it is now widely used in control theory.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0200587
Program Officer
Joe W. Jenkins
Project Start
Project End
Budget Start
2002-06-15
Budget End
2007-05-31
Support Year
Fiscal Year
2002
Total Cost
$98,094
Indirect Cost
Name
University of Alabama Tuscaloosa
Department
Type
DUNS #
City
Tuscaloosa
State
AL
Country
United States
Zip Code
35487