An operator corona theorem

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 D, the condition F* (z)F(z) ≥ δ 2 I ∀ 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 = F 0 + F 1 , where F 0 is a bounded analytic operator-valued function with a bounded analytic left in verse, and the Hilbert-Schmidt norms of operators F 1 (z) are uniformly (in z E D) bounded. It is now well-known that without any additional assumption, the condition F*F > δ 2 I 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 functions such that the operators F(z) act from a finite-dimensional space, but the general case is new.