Ideals in big Lipschitz algebras of analytic functions

• Autorzy: Thomas Vils Pedersen
• Języki publikacji: EN

Abstrakty

EN

For 0 < γ ≤ 1, let $Λ⁺_{γ}$ be the big Lipschitz algebra of functions analytic on the open unit disc 𝔻 which satisfy a Lipschitz condition of order γ on 𝔻̅. For a closed set E on the unit circle 𝕋 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 ∈ 𝕋: 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 ⊆ 𝕋 and some inner function Q.

Kategorie tematyczne

• 46J20: Ideals, maximal ideals, boundaries
• 30H05: Bounded analytic functions
• 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: 161
• Numer: 1
• Strony: 33-59

Daty

wydano

2004

Twórcy

autor

Thomas Vils Pedersen

• Department of Mathematics and Physics, The Royal Veterinary and Agricultural University, Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark

Identyfikatory

DOI

10.4064/sm161-1-3