Hilbert spaces of analytic functions and their applications

Clemson University, Clemson SC

Investigators

Abstract

Most of the proposed research activities in this project can be viewed essentially as perturbation problems for contractive operator semigroups. The proposed treatment of these questions can be briefly described as follows. One represents the Hilbert space on which the contractive semigroup acts as a chain of subspaces so that on each of the subspaces the semigroup acts almost as a group of unitaries. The corresponding spectral picture produces a chain of spaces of analytic functions. The goal is then to understand the properties of the semigroup by investigating these spaces and their relationship to the spaces generated by a certain model semigroup. The latter often reduces to a problem about invertibility properties of certain Toeplitz operators. In its essence, the method that the investigator suggests to treat these problems can be viewed as a delicate form of the classical argument principle. It has in its basis the powerful method of N. Makarov and A. Poltoratski regarding the injectivity problem for Toeplitz operators. As is often the case in harmonic analysis, the delicate properties of the Hilbert transform again should play the central role. The majority of the problems proposed in this project lie within the area of harmonic analysis. The core idea of harmonic analysis is the possibility of representing complicated signals as a combination of simpler signals - atoms which are in a sense canonical for the problem at hand. Due to the imprecision of measurements one is sometimes required to use a slightly different system of atoms, which may or may not possess the ideal properties of the canonical system. Careful analysis is required to determine to which extent the properties present under ideal measurements continue to hold in a real world situation. A large part of this project is devoted to the further development of the mathematical tools necessary for such an analysis. Potential applications are possible in the areas of signal processing and control theory. In addition, another goal of this project is to popularize the classical areas of harmonic and complex analysis by softening some existing deep techniques, thus making them more accessible to the future generations of mathematicians as well as to other scientists. As a member of a major science-technology university, the investigator will also incorporate some of these new ideas to offer an up-to-date, quality education of the new generations of STEM majors, at both the undergraduate and graduate level.