Summary: 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 \[ \left(S_\varphi f\right)(z)=\int_{\Pi}f(w)\varphi(z-\overline{w}) d\mu(w),\;\forall f\in\mathcal{A}^2(\Pi),\,z\in \Pi, \] where \(\varphi\) is an analytic function on \(\Pi\) given by \[ \varphi(z)=\int_{\mathbb{R}_+}\xi\sigma(\xi)e^{iz\xi}d\xi, \; \forall z\in\Pi \] 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 \(\mathbf{a}\in L^\infty(\Pi)\).