Common zero sets of equivalent singular inner functions

Keiji Izuchi¹

¹ Department of Mathematics Niigata University Niigata 950-2181, Japan

Studia Mathematica, Tome 163 (2004) no. 3, pp. 231-255

Résumé

Let $\mu $ and $\lambda$ be bounded positive singular measures on the unit circle such that $\mu \perp \lambda$. It is proved that there exist positive measures $\mu_0$ and $\lambda_0$ such that $\mu_0 \sim \mu$, $\lambda_0 \sim \lambda$, and $\{|\psi_{\mu_0}| 1\} \cap \{|\psi_{\lambda_0}| 1\} = \emptyset$, where $\psi_\mu$ is the associated singular inner function of $\mu$. Let ${\cal Z}(\mu) = \bigcap_{\{\nu;\,\nu \sim \mu\}} Z(\psi_\nu)$ be the common zeros of equivalent singular inner functions of $\psi_\mu$. Then ${\cal Z}(\mu) \not= \emptyset$ and ${\cal Z}(\mu) \cap {\cal Z}(\lambda) = \emptyset$. It follows that $\mu \ll \lambda$ if and only if ${\cal Z}(\mu) \subset {\cal Z}(\lambda)$. Hence ${\cal Z}(\mu)$ is the set in the maximal ideal space of $H^\infty$ which relates naturally to the set of measures equivalent to $\mu$. Some basic properties of ${\cal Z}(\mu)$ are given.

Détail

DOI : 10.4064/sm163-3-3

Keywords: lambda bounded positive singular measures unit circle perp lambda proved there exist positive measures lambda sim lambda sim lambda psi cap psi lambda emptyset where psi associated singular inner function nbsp cal bigcap sim psi common zeros equivalent singular inner functions psi cal emptyset cal cap cal lambda emptyset follows lambda only cal subset cal lambda hence cal set maximal ideal space infty which relates naturally set measures equivalent basic properties cal given