Spectral Theory of Generalized Toeplitz Operators

Introduction to Spectral Theory of Hankel and Toeplitz Operators

These are the notes of the lecture course given at LTCC in 2015. The aim of the course is to consider the following three classes of operators: Toeplitz and Hankel operators on the Hardy space on the unit circle and Toeplitz operators on the Bergman space on the unit disk. For each of these three classes of operators, we consider the following questions: boundedness and estimates or explicit ex...

Toeplitz operators in TQFT via skein theory

Topological quantum field theory associates to a punctured surface Σ, a level r and colors c in {1, . . . , r− 1} at the marked points a finite dimensional hermitian space Vr(Σ, c). Curves γ on Σ act as Hermitian operator T γ r on these spaces. In the case of the punctured torus and the 4 times punctured sphere, we prove that the matrix elements of T γ r have an asymptotic expansion in powers o...

Representation and Index Theory for Toeplitz Operators

We study Toeplitz operators on the Hardy spaces of connected compact abelian groups and of tube-type bounded symmetric domains. A representation theorem for these operators and for classes of abstract Toeplitz elements in C*-algebras is proved. This is used to give a unified treatment to index theory in this setting, and a variety of new index theorems are proved that generalize the Gohberg–Kre...

Spectral theory of Toeplitz and Hankel operators on the Bergman space A^1

The Fredholm properties of Toeplitz operators on the Bergman space A have been well-known for continuous symbols since the 1970s. We investigate the case p = 1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on A that ari...

Generalized Locally Toeplitz Sequences: Theory and Applications