Norm controlled inversions and a corona theorem for H∞-quotient algebras

Let Θ be an inner function on the unit disc D. We give a description of those Θ for which the quotient algebra H∞/ΘH∞ has no corona with respect to the visible part of its spectrum, that is for which M(H∞/ΘH∞) = {z ∈ D: Θ(z) = 0}M(H∞). It happens that this property is equivalent to the norm controlled inversion property for H∞/ΘH∞, as well as to a kind of weakened Carleson type embedding theorem. The quotient algebra A(D)/ΘH∞ is also considered. An interpretation of our main results in terms of model operators is given, too. © 2008 Elsevier Inc. All rights reserved