Multiplication operators on $S^2(\mathbb{D})$

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.