  In this paper, we study the multiplication operators on $S^2$, the space of
analytic functions on the open unit disk $\mathbb D$ whose first derivative is
in $H^2$. Specifically, we characterize the bounded and the compact
multiplication operators, establish estimates on the operator norm, and
determine the spectrum. Finally, we prove that the isometric multiplication
operators are precisely those induced by a constant function of modulus one.
