An Operator Corona Theorem for Some Subspaces of H^∞

Abstract.

Let E, E_* be separable Hilbert spaces. If S is an open subset of ${\mathbb{T}}$, then $A_S({\mathcal{L}}(E, E_{*}))$ denotes the space of all functions $f : {\mathbb{D}} \cup S \rightarrow {\mathcal{L}}(E, E_{*})$ that are holomorphic in $\mathbb{D}$, and bounded and continuous on $\mathbb{D} \cup S$. In this article we prove the following results:

• 1. A theorem concerning the approximation of $f \in A_S({\mathcal{L}}(E, E_{*}))$ by a function F that is holomorphic in a neighbourhood of $\mathbb{D} \cup S$ and such that the error F − f is uniformly bounded in the disk $\mathbb{D}$.

• 2. The corona theorem for $A_S({\mathcal{L}}(E, E_{*}))$ when dim(E) < ∞: If there exists a δ > 0 such that for all $z \in {\mathbb{D}} \cup S$, $f(z)^{*}f(z) \geq \delta^{2}I$, then there exists a $g \in A_S({\mathcal{L}}(E_{*}, E))$ such that for all $z \in {\mathbb{D}} \cup S$, g(z)f(z) = I.

• 3. The problem of complementing to an isomorphism for $A_S({\mathcal{L}}(E, E_{*}))$ when {dim(E) < ∞ (Tolokonnikov’s lemma): $f \in A_S({\mathcal{L}}(E, E_{*}))$ has a left inverse $g \in A_S({\mathcal{L}}(E_{*}, E))$ iff it is a ‘part’ of an invertible element F in $A_S({\mathcal{L}}(E_{*}))$.