Common zero sets of equivalent singular inner functions

Keiji Izuchi. "Common zero sets of equivalent singular inner functions." Studia Mathematica 163.3 (2004): 231-255. <http://eudml.org/doc/286515>.

@article{KeijiIzuchi2004,
abstract = {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.},
author = {Keiji Izuchi},
journal = {Studia Mathematica},
keywords = {maximal ideal space; zero set; singular inner function},
language = {eng},
number = {3},
pages = {231-255},
title = {Common zero sets of equivalent singular inner functions},
url = {http://eudml.org/doc/286515},
volume = {163},
year = {2004},
}

TY - JOUR
AU - Keiji Izuchi
TI - Common zero sets of equivalent singular inner functions
JO - Studia Mathematica
PY - 2004
VL - 163
IS - 3
SP - 231
EP - 255
AB - 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.
LA - eng
KW - maximal ideal space; zero set; singular inner function
UR - http://eudml.org/doc/286515
ER -