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. 
Jos\'e \'Angel Pel\'aez, Jouni R\"atty\"a and Kian Sierra
math.FA math.CV