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Closed ideals of certain Beurling algebras and application to operators with countable spectrum. (Idéaux fermés de certaines algèbres de Beurling et application aux opérateurs à spectre dénombrable.) (French. English summary) Zbl 1081.46033

This paper deals with generalizations of results of A. Atzmon [Acta Math. 144, 27–63 (1980; Zbl 0449.47007)] and of C. Bennett and J. E. Gilbert [Ann. Inst. Fourier (Grenoble) 22, 1–19 (1972; Zbl 0228.46046)] characterizing certain closed ideals in the Banach algebra \(A_{\omega }\) of all continuous complex-valued functions \(f\) on the unit circle \({\mathbb T} \) such that \( \sum_{n=-\infty }^{\infty }| \widehat{f}( n) | \omega ( n) \) converges. Here, \(\omega \) is a weight function such that \( \omega (n)\) is equal to \(( 1+n) ^{s}\) for natural numbers \(n\) (and a given positive number \(s)\) satisfying certain growth conditions for negative integers. For an ideal \(I\) and a natural number \(k\leq s\), define \( h^{k}(I)\) as the set of all \(z\in {\mathbb T} \) such that \(f^{( j) }(z)=0\) for all \(f\in I\) and \(j=0,\dots,k.\)
The first main result states that each closed ideal \(I\) in \(A_{\omega }\) such that \(h^{0}(I)\) is countable is equal to the space of all \(f\in A_{\omega }\) with \(f^{( j) }(z)=0\) for all \(z\in h^{j}(I),\) \(j=0,\dots,[ s] ,\) where \([ s] \) is the entire part of \(s.\) A similar result is proven for the Banach algebra \( A_{\omega }^{+}\) consisting of all \(f\in \) \(A_{\omega }\) with vanishing Fourier coefficients \(\widehat{f}(n)\) for negative integers \(n.\) The author uses these results to prove the following fact in spectral theory: Let \(E\) be a closed countable subset of \({\mathbb T} \) satisfying the condition of Carleson; then \(E\) is an interpolating set for \({\mathcal A}^{\infty }\), the space of infinitely differentiable functions in the closed unit disk and holomorphic in the open unit disk, if and only if for each positive number \(s \) there exists a number \(t\geq s\) such that \(\| T^{-n}\| \leq O(n^{t})\) for \(n\to \infty \) whenever \(T\) is a bounded invertible operator on some Banach space with the following properties: (i) the spectrum is contained in \(E,\) (ii) \(\| T^{n}\| \leq O(n^{s}) \), \( n\to \infty ,\) and (iii) \(\| T^{-n}\| =O(e^{\varepsilon \sqrt{n}})\), \(n\to \infty \), for each \(\varepsilon >0.\)

MSC:

46J20 Ideals, maximal ideals, boundaries
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
30H05 Spaces of bounded analytic functions of one complex variable
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