An operator corona theorem for a class of subspaces of H

Let E, E∗ be separable Hilbert spaces. If S is an open subset of T, then AS(L (E, E∗)) denotes the space of all functions f : D ∪ S → L (E, E∗) that are holomorphic in D, and bounded and continuous on D ∪ S. In this article we prove the following main results: 1. A theorem concerning the approximation of f ∈ AS(L (E, E∗)) by a function F that is holomorphic in a neighbourhood of D∪S and such that the error F −f is uniformly bounded in the disk D. 2. The corona theorem for AS(L (E, E∗)) when dim(E) < ∞: If there exists a δ > 0 such that for all z ∈ D ∪ S, f(z)f(z) ≥ δI, then there exists a g ∈ AS(L (E∗, E)) such that for all z ∈ D ∪ S, g(z)f(z) = I. 3. The problem of complementing to an isomorphism for AS(L (E, E∗)) when dim(E) < ∞ (Tolokonnikov’s lemma): f ∈ AS(L (E, E∗)) has a left inverse g ∈ AS(L (E∗, E)) iff it is a ‘part’ of an invertible element F in AS(L (E∗)). 4. A corona theorem for A(L (E, E∗)) when dim(E) = ∞, and the corona data function f is a ‘small’ perturbation of a ‘nice’ function f0. MSC numbers: 30H05 (primary), 46J15, 47A56 (secondary)