Operator Theory

Operator theory is no doubt a diverse area that has grown out of linear algebra and complex analysis, and is often described as the branch of functional analysis that deals with bounded linear operators and their (spectral) properties. It has developed with strong links to (mathematical) physics and mechanics, and continues to attract both pure and applied mathematicians in its vast area.

We start by quickly reviewing the core material in functional analysis, which should provide (with some extra effort) a sufficient background for those with no previous experience in functional analysis. This is followed by the study of abstract Banach algebras and spectral theory. Linear operators are considered both on Hilbert spaces and Banach spaces, and in particular we focus on compact and Fredholm operators and their index theory.

In order to study certain classes of concrete linear operators on analytic function spaces, we develop the theory of Hardy and Bergman spaces, and study spectral properties of these operators with tools introduced in the more abstract setting. This part of the course reveals the fruitful interplay between comples analysis and operator theory.

As pointed above, applications play an important role in connection with operator theory, and so we introduce as an illustration some of the most important of these, namely orthogonal polynomials and random matrices, which have received much attention lately.

In summary, we focus on the following topics:

•     review of the necessary preliminaries on functional analysis
•     Banach algebra techniques
•     compact and Fredholm operators
•     analytic function spaces
•     concrete classes of linear operators
•     the use of tools and techniques illustrated with Hankel and Toeplitz operators
•     applications: orthogonal polynomials and random matrices

The course is suitable for those interested in analysis, mathematical physics, or applied mathematics, and provides many topics ideal for further research, e.g. for Master's thesis and beyond. This can be further discussed with the lecturer and also with other members of the analysis group.

Textbooks:

1. Ronald G. Douglas, Banach algebra techniques in operator theory
2. Kehe Zhu, Operator theory in function spaces
3. Walter Rudin, Functional Analysis