Berezin transform and Toeplitz operators on weighted 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 $${\mathcal{T}}_{\mu }(f)(z)={\int }_{\mathbb{D}}f(\zeta )\overline{B_z^{\omega }(\zeta )}d\mu (\zeta ),$$ where $B_z^{\omega }$ are the reproducing kernels of the weighted Bergman space $A_{\omega }^2$. We describe bounded and compact Toeplitz operators ${\mathcal{T}}_{\mu }:A_{\omega }^p\to A_{\omega }^q$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$\widetilde{{\mathcal{T}}_{\mu }}(z)=\frac{⟨{\mathcal{T}}_{\mu }(B_z^{\omega }),B_z^{\omega }{⟩}_{A_{\omega }^2}}{\Vert B_z^{\omega }{\Vert }_{A_{\omega }^2}^2}.$$ We also characterize Schatten class Toeplitz operators in terms of the Berezin transform and apply this result to study Schatten class composition operators.

References in corpus (1)

• Embedding theorems for Bergman spaces via harmonic analysis