Integral representation of vertical operators on the Bergman space over the upper half-plane

Abstract

Let $\Pi $ denote the upper half-plane. In this article, we prove that every vertical operator on the Bergman space $\mathcal{A}^2(\Pi )$ over the upper half-plane can be uniquely represented as an integral operator of the form \begin{equation*} \left(S_\varphi f\right)(z) = \int _{\Pi } f(w) \varphi (z-\overline{w}) d\mu (w),~~\forall f\in \mathcal{A}^2(\Pi ),~z\in \Pi , \end{equation*} where $\varphi $ is an analytic function on $\Pi $ given by \begin{equation*} \varphi (z) = \int _{\mathbb{R}_+}\xi \sigma (\xi ) e^{iz\xi } d\xi , \ \forall z\in \Pi \end{equation*} for some $\sigma \in L^\infty (\mathbb{R}_+)$. Here $d\mu (w)$ is the Lebesgue measure on $\Pi $. Later on, with the help of above integral representation, we obtain various operator theoretic properties of the vertical operators.Also, we give integral representation of the form $S_\varphi $ for all the operators in the $C^\ast $-algebra generated by Toeplitz operators $T_{\mathbf{a}}$ with vertical symbols ${\mathbf{a}}\in L^\infty (\Pi )$.

Keywords

• Bergman space
• multiplication operator
• reducing subspace
• Toeplitz operator