Journal of the
Korean Mathematical Society JKMS

Let $\mathcal{M}$ be the maximal ideal space of the Banach algebra $H^\infty$ of bounded analytic functions on the open unit disc $\Delta$. For a positive singular measure $\mu$ on $\partial \Delta$, let $L^1_+(\mu)$ be the set of measures $\nu$ with $0 \le \nu \ll \mu$ and $\psi_\nu$ the associated singular inner functions. Let $\mathcal{R}(\mu)$ and $\mathcal{R}_0(\mu)$ be the union sets of $\{|\psi_\nu| < 1\}$ and $\{\psi_\nu = 0\}$ in $\mathcal{M} \setminus \Delta$, $\nu \in L^1_+(\mu)$, respectively. It is proved that if $S(\mu) = \partial \Delta$, where $S(\mu)$ is the closed support set of $\mu$, then $\mathcal{R}(\mu) = \mathcal{R}_0(\mu) = \mathcal{M} \setminus (\Delta \cup M(L^\infty(\partial\Delta)))$ and $L^\infty(\partial\Delta)$ is generated by $H^\infty$ and $\overline{\psi_\nu}, \nu \in L^1_+(\mu)$. It is proved that $d\theta(S(\mu)) = 0$ if and only if there exists a Blaschke product $b$ with zeros $\{z_n\}_n$ such that $\mathcal{R}(\mu) \subset \{|b| < 1\}$ and $S(\mu)$ coincides with the set of cluster points of $\{z_n\}_n$. While, we prove that $\mu$ is a sum of finitely many point measures if and only if there exists another positive singular measure $\lambda$ such that $\mathcal{R}(\mu) \subset \{|\psi_\lambda| < 1\}$ and $S(\lambda) = S(\mu)$. Also it is studied conditions on $\mu$ for which $\mathcal{R}(\mu) = \mathcal{R}_0(\mu)$.

Keywords: singular inner function, bounded analytic function, maximal ideal space

MSC numbers: Primary 46J15