This course lies at the interface between complex analysis and operator theory. It will introduce the classical Hardy spaces, together with some of their cousins, and present the Toeplitz and Hankel operators defined on them. Applications to approximation and interpolation will be given.
Familiarity with the main theorems of elementary complex analysis. Some
experience of Hilbert spaces and the concept of a bounded linear operator.
The definition, at least, of a Banach space.
This course will include most of the following:
1. Introduction. Examples of such spaces (Hardy spaces, Bergman spaces,
Wiener algebra, Paley-Wiener space). (1)
2. Hardy spaces on the disc. Poisson kernel. Inner and outer functions. (5)
3. Operators on H
2 and L
2. Laurent, Toeplitz and Hankel operators.
Nehari, Carathéodory-Fejér and Nevanlinna-Pick problems. Hilbert
transform. (4)