Functional analysis and operator theory

The study of various spaces of analytic functions is a topic active ongoing research. The most common spaces are different variants of Hardy, Bergman, BMOA and Bloch spaces. The basic theory of all of these provides good topics for thesis projects that can be adapted to the student's interests and ambitions. There are also many natural concrete operators acting on these spaces, such as Toeplitz, Hankel, Volterra and composition operators.

Possible topics:

• Basic theory of Hardy and/or Bergman spaces
• Bergman kernel and Bergman projection
• Bloch space and conformal mappings
• Theory of one or several concrete operators on these spaces
• Carleson measures

It is also possible to make project of more functional analytic nature. These projects do not necessarily need to be linked to complex analysis or operator theory.

Possible topics:

• Spectral theory and Fredholm operators
• Topological vector spaces
• Zorn's lemma in analysis

Other topics can also be discussed. Many of these topics are quite advanced for a BSc project; instead it possible to choose one these topics and study the basic theory needed for dealing with these.

Suggested literature (note that all of these books go way beyond the scope of a BSc/Msc project):

• Zhu, Operator theory in function spaces
• Zhu, Spaces of holomorphic functions in the unit ball
• Böttcher and Silbermann, Analysis of Toeplitz operators
• Pavlovic, Function classes in the unit disk
• Rudin, Functional analysis
• Horvath, Topological vector spaces and distributions