Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen

Studia Mathematica, Tome 162 (2004), p. 33-59 / Harvested from The Polish Digital Mathematics Library

Résumé

For 0 < γ ≤ 1, let $Λ⁺_{γ}$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{γ}(E,Q)$ be the closed ideal in $Λ⁺_{γ}$ consisting of those functions $f ∈ Λ⁺_{γ}$ for which (i) f = 0 on E, (ii) $|f(z)-f(w)| = o(|z-w|^{γ})$ as d(z,E),d(w,E) → 0, (iii) $f/Q ∈ Λ⁺_{γ}$. Also, for a closed ideal I in $Λ⁺_{γ}$, let $E_{I}$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_{I}$ be the greatest common divisor of the inner parts of non-zero functions in I. Our main conjecture about the ideal structure in $Λ⁺_{γ}$ is that $J_{γ}(E_{I},Q_{I}) ⊆ I$ for every closed ideal I in $Λ⁺_{γ}$. We confirm the conjecture for closed ideals I in $Λ⁺_{γ}$ for which $E_{I}$ is countable and obtain partial results in the case where $Q_{I} = 1$. Moreover, we show that every wk* closed ideal in $Λ⁺_{γ}$ is of the form f ∈ $Λ⁺_{γ}$: f = 0 on E and f/Q ∈ $Λ⁺_{γ}$ for some closed set E ⊆ and some inner function Q.

Détail

Publié le : 2004-01-01

Zbl 1054.46037

EUDML-ID : urn:eudml:doc:284948