Composition operators on the Dirichlet space of the disk. Lecture 1

The Dirichlet space D of the disk D, which is a Hilbert space of analytic functions on D, turns out to be fairly different from the Hardy space, and to be less well known (see yet the quite recent and excellent book by El-Fallah, Kellay, Mashreghi, Ransford). The purpose of this course will be to present some recent results concerning composition operators on that space, obtained in collaboration with P.Lefevre, D.Li, L.Rodriguez-Piazza.

Accordingly, the course will be divided in three parts:

1. Generalities on Hilbert spaces of analytic functions, reproducing kernel, Carleson measures, etc... Definition of the Dirichlet space. Logarithmic capacity and Beurling’s theorem on the existence of quasi-everywhere radial limits. Absence of a Littlewood subordination principle and intrinsic difficulties. Zorboska’s criterion.
2. Generalities on compact operators on a Hilbert space and their singular numbers. Approximation numbers, Bernstein, Gelfand, Kolmogorov numbers. Weyl’s multiplicative inequalities. Examples (in particular in the framework of Hankel or composition operators). The Megretskii-Peller-Treil theorem. State of art for the Hardy space.
3. Composition operators on the Dirichlet space and their approximation numbers. Peak sets for D and time of stay of a compact symbol at the boundary. Optimality of the Gallardo-Gonzalez result. Green capacity and spectral radius-type formula. Examples. Lower bound result of the Megretskii-Peller-Treil type.