Spectral theory of Toeplitz operators on Bergman spaces

Toeplitz operators play an extremely important role in operator theory and its applications. They form one of the most important classes of non-selfadjoint operators and their spectral theory is remarkably rich. The study of Toeplitz operators demonstrates interplay between function theory, functional analysis and linear algebra with plenty of application in other areas of mathematics. Toeplitz operators play an important role in many recent applications in engineering, physics, and mathematics physics. This project is concerned with the spectral properties of Toeplitz operators on Bergman spaces. Their theory in Hardy spaces is much better understood than in the context of Hardy spaces. For example, there are no complete characterizations of bounded or compact Toeplitz operators on Bergman spaces, and much less is known about their Fredholm properties in these spaces than in Hardy spaces. One of the main themes of the project is to study the essential spectra of Toeplitz operators with piecewise continuous symbols on Bergman spaces. In Hardy spaces with Muckenhoupt weights and Carleson curves, the study of piecewise continuous symbols is now complete (see "Carleson curves, Muckenhoupt weights, and Toeplitz operators" by Böttcher and Silbermann). Motivated by this beautiful theory, we propose a similar study of Toeplitz operators in Bergman spaces, where only the Hilbert space case has been settled. In addition to the essential spectra, we also aim to study other (spectral) properties of these operators on Bergman spaces.