Operator Theory On Spaces Of Analytic Functions

Timetable

Module aim (aims)

The aim of the course to present the modern theory of concrete operators on spaces of holomorphic functions. We intend to provide both some introduction to the theory of such spaces of holomorphic functions as the Hardy and the Bergman space and to the theory of operators acting on these spaces. In particular, Toeplitz and Hankel operators will be discussed. We shall start by presenting the classical theory of operators on the Hardy space, which will motive further study of the Bergman space case.
Operator theory on spaces of holomorphic functions is a beautiful, active area of research, which combines methods of complex analysis, functional analysis and (abstract) operator theory.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Although we shall discuss many rather up-to-date results in operator theory, the aim of the course is to be as self-contained as possible. Basic information in complex analysis, functional analysis and operator theory will be presented in the course. However some background in analytic functions will be helpful. A student may for example study Analytic Functions course at the same time.

Syllabus

Week 1: Holomorphic functions.

Week 2: Holomorphic functions.

Week 3: Hardy spaces in the unit disk.

Week 4: Hardy spaces in the unit disk.

Week 5: Bergman spaces in the unit disk.

Week 6: Bergman spaces in the unit disk.

Week 7: Banach spaces, Hilbert spaces, basic functional analysis.

Week 8: Abstract operator theory.

Week 9: Abstract operator theory.

Week 10: Toeplitz operators on the Hardy spaces.

Week 11: Toeplitz operators on the Hardy spaces.

Week 12: Hankel operators on the Hardy spaces

Week 13: Hankel operators on the Hardy spaces.

Week 14: Toeplitz operators on the Bergman spaces.

Week 15: Algebras of Toeplitz operators on the Bergman spaces.

Reading list

1. A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Berlin 1990.
2. V. Peller, Hankel Operators and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, New York 2003.
3. R. G. Douglas, Banach Algebra Techniques in Operator Theory, Pure and Applied Mathematics, Academic Press, New York 1973.
4. N. Nikolski, Hardy spaces, Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge 2019.
5. N. Nikolski, Toeplitz Matrices and Operators, Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge 2020.
6. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer 2000.
7. N. L. Vasilevsky, Commutative Algebras of Toeplitz Operators on the Bergman Space, Operator Theory Advances and Applications vol. 185, Birkhäuser, Basel, Boston, Berlin, 2008.