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

Abstract

Abstract. 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 algebraIf I is a closed ideal of A ω (T), we set h 0 (I) = {z ∈ T : f (z) = 0 (f ∈ I)}. We describe all closed ideals I of A ω (T) such that h 0 (I) is at most countable. A similar result is obtained for closed ideals of the algebra A + s (T) = {f ∈ A ω (T) : 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 a ∞ , the space of infinitely differentiable functions in the closed unit disc D and holomorphic in D.

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

“…Observe also that Theorem 1.2 is not valid when σ(T ) is a null measure set (see [10]). Similar results of Tauberian type were obtained in [1,2,5,6,7,8,10,14].…”

“…The conclusion follows from the proof of Theorem 4.1 of [5] (see also [2]) for Theorem 1.1, and from the proof of Theorem 6.4 of [6] for Theorem 1.2. Now suppose that condition (2) is satisfied for some α ∈ (0, 1).…”

Closed ideals with countable hull in algebras of analytic functions smooth up to the boundary

“…The structure of closed ideals I of A + s (T) such that h 0 (I) is countable, is known only in the cases when s < 1 (see [23], [3] and [7]) and when s ≥ 1 and U I = 1 (see [1]). Here we give a complete characterisation of such ideals.…”