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Abstract

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