Berezin Transform and Toeplitz Operators on Bergman Spaces Induced by Regular Weights

Abstract

Given a regular weight $\omega$ and a positive Borel measure $\mu$ on the unit disc $\mathbb {D}$, the Toeplitz operator associated with $\mu$ is

$$\begin{aligned} {\mathcal {T}}_\mu (f)(z)=\int _\mathbb {D}f(\zeta )\overline{B_z^\omega (\zeta )}\,\mathrm{d}\mu (\zeta ), \end{aligned}$$

where $B^\omega _{z}$ are the reproducing kernels of the weighted Bergman space $A^2_\omega$. The primary purpose of this paper is to study the interrelationships between the Toeplitz operator ${\mathcal {T}}_\mu$, Carleson measures, and the Berezin transform

$$\begin{aligned} \widetilde{{\mathcal {T}}}_\mu (z)=\frac{\langle {\mathcal {T}}_\mu (B^\omega _{z}), B^\omega _{z} \rangle _{A^2_\omega }}{\Vert B_z^\omega \Vert ^2_{A^2_\omega }}. \end{aligned}$$

We provide descriptions of bounded and compact operators ${\mathcal {T}}_\mu :A^p_\omega \rightarrow A^q_\omega$, $1<q,p<\infty$, as well as, Schatten class Toeplitz operators ${\mathcal {T}}_\mu$ on $A^2_\omega$. The last mentioned characterization is applied to study Schatten class composition operators on $A^2_\omega$.