We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the $n$-dimensional complex plane. Characterization of the commutant of such operators is given.

We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on the weighted Bergman space $A^2\left(\Omega \right)$, where $\Omega$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^p{\overline{z}}^q}$ to be complex symmetric. When $p,q\in {\mathbb{N}}^2$, we prove that $T_{z^p{\overline{z}}^q}$ is complex symmetric on $A^2\left(\Omega \right)$ if and only if $p_1=q_2$ and $p_2=q_1$. Moreover, we completely characterize when monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on $A^2\left({𝔻}_n\right)$ are $J_U$-symmetric with the $n\times n$ symmetric unitary matrix $U$.