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