Title: An operator corona theorem
Authors: Sergei Treil
Issue: Volume 53 (2004), Issue 6, 1765-1784
Abstract:
![In this paper some new positive results in the Operator Corona Problem are obtained in a rather general situation. The main result is that, under some additional assumptions about a bounded analytic operator-valued function $F$ in the unit disc $\mathbb{D}$, the condition $$F^{*}(z)F(z) \ge \delta^2I \ \forall z \in \mathbb{D}, (\delta>0)$$ implies that $F$ has a bounded analytic left inverse. Typical additional assumptions are (any of the following): egin{enumerate}[(1)] item The trace norms of defects $I-F^{*}(z)F(z)$ are uniformly (in $z \in \mathbb{D}$) bounded. The identity operator $I$ can be replaced by an arbitrary bounded operator here, and $F^{*}F$ can be changed to $FF^{*}$; \item The function $F$ can be represented as $F = F_0 + F_1$, where $F_0$ is a bounded analytic operator-valued function with a bounded analytic left inverse, and the Hilbert-Schmidt norms of operators $F_1(z)$ are uniformly (in $z \in \mathbb{D}$) bounded.end{enumerate} It is now well known that without any additional assumption, the condition $F^{*}F \ge \delta^2I$ is not sufficient for the existence of a bounded analytic left inverse.\par Another important result of the paper is the so-called Tolokonnikov's Lemma, which says that a bounded analytic operator-valued function has an analytic left inverse if and only if it can be represented as a 'part' of an invertible bounded analytic function. This result was known for operator-valued functions such that the operators $F(z)$ act from a finite-dimensional space, but the general case is new.](https://iumj.s3-us-west-2.amazonaws.com/abstracts/2640.png)
