Berezin transform and Toeplitz operators on weighted Bergman spaces induced by regular weights

TL;DR

This paper characterizes bounded, compact, and Schatten class Toeplitz operators on weighted Bergman spaces using Carleson measures and the Berezin transform, and applies these results to composition operators.

Contribution

It provides new characterizations of Toeplitz operators on weighted Bergman spaces in terms of Carleson measures and the Berezin transform, including Schatten class criteria.

Findings

Characterization of bounded and compact Toeplitz operators via Carleson measures and Berezin transform.

Schatten class criteria for Toeplitz operators in terms of Berezin transform.

Application of results to Schatten class composition operators.

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 $$ \mathcal{T}_\mu(f)(z)=\int_{\mathbb{D}} f(\zeta)\bar{B_z^\omega(\zeta)}\,d\mu(\zeta), $$ where $B^\omega_{z}$ are the reproducing kernels of the weighted Bergman space $A^2_\omega$. We describe bounded and compact Toeplitz operators $\mathcal{T}_\mu:A^p_\omega\to A^q_\omega$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$ \widetilde{\mathcal{T}_\mu}(z)=\frac{\langle\mathcal{T}_\mu(B^\omega_{z}), B^\omega_{z} \rangle_{A^2_\omega}}{\|B_z^\omega\|^2_{A^2_\omega}}. $$ We also characterize Schatten class Toeplitz operators in terms of the Berezin transform and apply this result to study Schatten class composition operators.