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

Abstract

In this paper, we study Bergman projection ${\mathbb{P}}_{\alpha ,\beta }$ and Toeplitz operators $T_{\phi }^{\alpha ,\beta }$ 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 $T_{\phi }^{\alpha ,\beta }$ 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-Poincaré metric.