Common zero sets of equivalent singular inner functions

Keiji Izuchi

Studia Mathematica, Tome 162 (2004), p. 231-255 / Harvested from The Polish Digital Mathematics Library

Résumé

Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and ${|ψ_{μ₀}| < 1} ∩ {|ψ_{λ₀}| < 1} = ∅$, where $ψ_{μ}$ is the associated singular inner function of μ. Let $(μ) = ⋂_{ν;ν∼ μ} Z(ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$. Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{∞}$ which relates naturally to the set of measures equivalent to μ. Some basic properties of (μ) are given.

Détail

Publié le : 2004-01-01

Zbl 1074.46034

EUDML-ID : urn:eudml:doc:286515