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In this paper, we study Bergman projection $\mathbb{P}_{\alpha,\beta}$ andToeplitz operators $T^{\alpha,\beta}_\varphi$ on the $\beta$-modified Bergmanspace $\mathcal{A}_{\alpha,\beta}^p$. We give some properties of$\mathbb{P}_{\alpha,\beta}$ and a necessary and sufficient condition for$T^{\alpha,\beta}_\varphi$ to be compact. We end with a characterization ofToeplitz operators via Carleson measures by introducing a new Bergman metricinherited by the Bergman kernel $\mathbb{K}_{\alpha,\beta}$ that will beequivalent to the classical Bergman-Poincar\'e metric.

Toeplitz operators via Carleson measures On $\beta$-modified Bergman Spaces