Common zero sets of equivalent singular inner functions

• Autorzy: Keiji Izuchi
• Języki publikacji: EN

Abstrakty

EN

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 $𝓩(μ) = ⋂_{ν;ν∼ μ} Z(ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$. Then 𝓩(μ) ≠ ∅ and 𝓩(μ) ∩ 𝓩(λ) = ∅. It follows that μ ≪ λ if and only if 𝓩(μ) ⊂ 𝓩(λ). Hence 𝓩(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 𝓩(μ) are given.

Kategorie tematyczne

• 46J15: Banach algebras of differentiable or analytic functions, H p -spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2004
• Tom: 163
• Numer: 3
• Strony: 231-255

Daty

wydano

2004

Twórcy

autor

Keiji Izuchi

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

Identyfikatory

DOI

10.4064/sm163-3-3