主题：系列讲座预告

On the Hardy space, Toeplitz operators and Hankel operators are fascinating examples of the fruitful interplay between operator theory, operator algebras, function theory, harmonic analysis, partial differential equations and several complex analysis. Moreover, those operators are of importance in applied mathematics such as system theory, and stationary stochastic processes.

The operator theory on Bergman spaces is a masterful blend of complex function theory with functional analysis and operator theory. For the Bergman spaces of the unit disk, many new phenomena occur, new theorems and proofs had to be developed, and some basic questions are still not completely settled.

The research on Toeplitz operators on Bergman spaces has been quite active recently. Problems posed in Hardy spaces often make perfect sense on the Bergman space. But the techniques required to solve problems in this setting are very different from those that work in the Hardy spaces setting. However, one often sees similarities in the theorems (but not the proofs, although in both cases the proofs usually feature an interplay between function theory and operator theory).

In my lectures I will start basic materials in functional analysis and complex analysis and cover the following topics:
  Banach spaces, Lp spaces;
  Hilbert spaces;
  Bounded linear operators on Hilbert spaces;
  Bergman spaces and reproducing kernels;
  Dual of the Bergman spaces;
  Berezin transform;
  Toeplitz operators;
  Hankel operators.

The most materials covered in my lectures are contained in
Axler, Sheldon, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, 1–50, Pitman Res. Notes Math. Ser., 171, Longman Sci. Tech., Harlow, 1988.
Zhu, Kehe, Operator theory in function spaces, Second edition, Mathematical Surveys and Monographs, 138. American Mathematical Society, Providence, RI, 2007. xvi+348 pp.