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Idéaux fermés de certaines algèbres de Beurling et application aux opérateurs à spectre dénombrable

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Agrafeuil C.

Studia Mathematica

2005

tom 167

nr 2

133-151

We denote by 𝕋 the unit circle and by 𝔻 the unit disc of ℂ. Let s be a non-negative real and ω a weight such that $ω(n) = (1 + n)^{s}$ (n ≥ 0) and the sequence $(ω(-n)/(1+n)^{s})_{n≥0}$ is non-decreasing. We define the Banach algebra $A_{ω}(𝕋) = {f ∈ 𝓒(𝕋): ||f||_{ω} = ∑_{n=-∞}^{+∞} |f̂(n)|ω(n) < +∞ }$. If I is a closed ideal of $A_{ω}(𝕋)$, we set $h⁰(I) = {z ∈ 𝕋: f(z) = 0 (f ∈ I)}$. We describe all closed ideals I of $A_{ω}(𝕋)$ such that h⁰(I) is at most countable. A similar result is obtained for closed ideals of the algebra $A⁺_{s}(𝕋) = {f ∈ A_{ω}(𝕋): f̂(n) = 0 (n < 0)}$ without inner factor. Then we use this description to establish a link between operators with countable spectrum and interpolating sets for $𝑎^{∞}$, the space of infinitely differentiable functions in the closed unit disc 𝔻̅ and holomorphic in 𝔻.

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1 Studia Mathematica

1 Agrafeuil C.

1 2005

Wyniki wyszukiwania

Idéaux fermés de certaines algèbres de Beurling et application aux opérateurs à spectre dénombrable