  Let $\Omega\subset \mathbb{C}^n$ be a smooth bounded pseudoconvex domain and
$A^2 (\Omega)$ denote its Bergman space. Let $P:L^2(\Omega)\longrightarrow
A^2(\Omega)$ be the Bergman projection. For a measurable
$\varphi:\Omega\longrightarrow \Omega$, the projected composition operator is
defined by $(K_\varphi f)(z) = P(f \circ \varphi)(z), z \in\Omega, f\in A^2
(\Omega).$ In 1994, Rochberg studied boundedness of $K_\varphi$ on the Hardy
space of the unit disk and obtained different necessary or sufficient
conditions for boundedness of $K_\varphi$. In this paper we are interested in
projected composition operators on Bergman spaces on pseudoconvex domains. We
study boundedness of this operator under the smoothness assumptions on the
symbol $\varphi$ on $\overline\Omega$.
