The Forelli problem concerning ideals in the disk algebra $A(\textbf {D})$

Abstract:

Let $Z(f)$ be the zero set of a function $f \in A({\mathbf {D}})$ and $Z(I) = { \cap _{f \in I}}Z(f)$ the zero set of an ideal $I$ in $A({\mathbf {D}})$. It is shown that in the disk algebra $A({\mathbf {D}})$ every finitely generated ideal $I$ has the weak Forelli property, i.e. there exists a function $f \in I$ such that $Z(f) \cap T = Z(I) \cap T$, where $T$ is the boundary of the unit circle ${\mathbf {D}}$. On the other hand, there exists a finitely generated ideal $I$ in $A({\mathbf {D}})$ such that $Z(f) \ne Z(I)$ for each choice of $f \in I$. This provides us with a negative answer to a problem of F. Forelli [1].