Closed ideas of certain Beurling algebras and application to operators at a countable spectrum

We denote by T the unit circle and by D the unit disc of C. Let s be a non-negative real and omega a weight such that omega(n) = (1 + n)(s) (n >= 0) and the sequence (omega(+ n)/(1 + n)(s))(n >= 0) is non-decreasing. We define the Banach algebra [GRAPHICS] If I is a closed ideal of A(omega)(T), we set h(0)(I) = {z epsilon T : f (z) = 0 (f epsilon I)}. We describe all closed ideals I of A(omega)(T) such that h(0)(1) is at most countable. A similar result is obtained for closed ideals of the algebra A(s)(+) (T) = {f epsilon A(omega)(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(infinity), the space of infinitely differentiable functions in the closed unit disc D and holomorphic in D.