The Berezin transform and operators on spaces of analytic functions

Abstract

In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6. Sarason's example 1. Spaces of analytic functions. In this section we will introduce the spaces of analytic functions on which we will be working. We start with the following general definition. Definition 1.1. A reproducing functional Hilbert space on an open subset Ω of C is a Hilbert space H of functions on Ω such that for every w ∈ Ω the linear functional f → f (w) is bounded on H. If H is a reproducing functional Hilbert space on set Ω, then by the Riesz Representation Theorem for every w ∈ Ω there is a unique element K w ∈ H for which f (w) = f, K w , for all f ∈ H. We call the function K w the reproducing kernel at w. Before we turn to a few examples we will prove some simple results about these reproducing kernels. The following proposition gives a way to compute the reproducing kernels.