Given a regular weight ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} and a positive Borel measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} on the unit disc D\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {D}$$\end{document}, the Toeplitz operator associated with μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} is Tμ(f)(z)=∫Df(ζ)Bzω(ζ)¯dμ(ζ),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {T}}_\mu (f)(z)=\int _\mathbb {D}f(\zeta )\overline{B_z^\omega (\zeta )}\,\mathrm{d}\mu (\zeta ), \end{aligned}$$\end{document}where Bzω\documentclass[12pt]{minimal}
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\begin{document}$$B^\omega _{z}$$\end{document} are the reproducing kernels of the weighted Bergman space Aω2\documentclass[12pt]{minimal}
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\begin{document}$$A^2_\omega $$\end{document}. The primary purpose of this paper is to study the interrelationships between the Toeplitz operator Tμ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {T}}_\mu $$\end{document}, Carleson measures, and the Berezin transform T~μ(z)=⟨Tμ(Bzω),Bzω⟩Aω2‖Bzω‖Aω22.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \widetilde{{\mathcal {T}}}_\mu (z)=\frac{\langle {\mathcal {T}}_\mu (B^\omega _{z}), B^\omega _{z} \rangle _{A^2_\omega }}{\Vert B_z^\omega \Vert ^2_{A^2_\omega }}. \end{aligned}$$\end{document}We provide descriptions of bounded and compact operators Tμ:Aωp→Aωq\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {T}}_\mu :A^p_\omega \rightarrow A^q_\omega $$\end{document}, 1<q,p<∞\documentclass[12pt]{minimal}
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\begin{document}$$1<q,p<\infty $$\end{document}, as well as, Schatten class Toeplitz operators Tμ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {T}}_\mu $$\end{document} on Aω2\documentclass[12pt]{minimal}
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\begin{document}$$A^2_\omega $$\end{document}. The last mentioned characterization is applied to study Schatten class composition operators on Aω2\documentclass[12pt]{minimal}
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\begin{document}$$A^2_\omega $$\end{document}.
