An Operator Corona Theorem

In this paper some new positive results in the Operator Corona Problem are obtained in rather general situation. The main result is that under some additional assumptions about a bounded analytic operator-valued function F in the unit disc D the condition F (z)F (z) ≥ δI ∀z ∈ D (δ > 0) implies that F has a bounded analytic left inverse. Typical additional assumptions are (any of the following): (1) The trace norms of defects I−F ∗(z)F (z) are uniformly (in z ∈ D) bounded. The identity operator I can be replaced by an arbitrary bounded operator here, and F ∗F can be changed to FF ∗; (2) The function F can be represented as F = F0 + F1, where F0 is a bounded analytic operator-valued function with a bounded analytic left inverse, and the Hilbert–Schmidt norms of operators F1(z) are uniformly (in z ∈ D) bounded. It is now well-known that without any additional assumption, the condition F ∗F ≥ δI is not sufficient for the existence of a bounded analytic left inverse. 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 function such that the operators F (z) act from a finitedimensional space, but the general case is new.