**Description**

The Hardy space is a famous model for the Hilbert space l_2 as holomorphic functions on the unit disk. It was introduced by Hardy almost 100 years ago. Since then, this space and the operators on it that can be related to analytic functions (Toeplitz and Hankel operators), have attracted the attention of many mathematicians, because of the beautiful intertwining of complex analysis, functional analysis and measure theory one encounters here, and also because of the applications that the theory has in engineering. In this course, we will concentrate on Toeplitz operators and the aesthetic aspect.

By taking this course you will learn more about complex function theory, see some functional analytic operator theory “in action”, and hopefully marvel at the interplay between complex function theory, functional analysis and measure theory that becomes visible when, e.g., studying invariant subspaces and spectra of Toeplitz operators on the Hardy space.

If you like abstract analysis, you are bound to like this course.

**Prerequisites**

Basic complex analysis, introduction to functional analysis, basic measure theory.

**Literature**

R.A. Martinez-Avendano and P. Rosenthal: “An Introduction to Operators on the Hardy-Hilbert Space”, ISBN: 978-0-387-35418-7, Springer, 2007 (47.65 euro).

**Examination**

Homework assignments only

**Remarks**

In principle, homework will be assigned biweekly, with a deadline three weeks after the assignment. Precise arrangements will depend on the schedule of the lectures.

PLEASE NOTE: Since low grades can be compensated by extra effort for the other assignments, there is no retake for this course.

This is a common course between Leiden and Delft, where it is known as “Advanced topics in analysis”. In Leiden this course (level 400) is eligible for both a Bachelor’s and a Master’s degree.

We may decide to use both Leiden and Delft as venue for the lectures, in consultation with the participants. See the lecturer’s home page for this, which we will also use as web base for this course.