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Some results about Beurling algebras with applications to operator theory

100%

Vils Pedersen T.

Studia Mathematica

1995

tom 115

nr 1

39-52

We prove that certain maximal ideals in Beurling algebras on the unit disc have approximate identities, and show the existence of functions with certain properties in these maximal ideals. We then use these results to prove that if T is a bounded operator on a Banach space X satisfying $∥T^n∥ = O(n^β)$ as n → ∞ for some β ≥ 0, then $∑_{n=1}^∞ ∥(1-T)^n x∥/∥(1-T)^{n-1}x∥$ diverges for every x ∈ X such that $(1-T)^{[β]+1}x ≠ 0$.

Ideals in big Lipschitz algebras of analytic functions

100%

Vils Pedersen T.

Studia Mathematica

2004

tom 161

nr 1

33-59

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.

Ograniczanie wyników

2 Studia Mathematica

2 Vils Pedersen T.

1 2004

1 1995

Wyniki wyszukiwania

Some results about Beurling algebras with applications to operator theory

Ideals in big Lipschitz algebras of analytic functions