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

In this paper, we study Bergman projection $\mathbb{P}_{\alpha,\beta}$ and Toeplitz operators $T^{\alpha,\beta}_\varphi$ on the $\beta$-modified Bergman space $\mathcal{A}_{\alpha,\beta}^p$. 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 of Toeplitz operators via Carleson measures by introducing a new Bergman metric inherited by the Bergman kernel $\mathbb{K}_{\alpha,\beta}$ that will be equivalent to the classical Bergman-Poincar\'e metric.