Toeplitz operators via Carleson measures On $β$-modified Bergman Spaces

Abstract: 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.