Notes on the Codimension One Conjecture in the Operator Corona Theorem

Abstract

Answering a question of S. R. Treil (2004), for every δ, 0 < δ < 1, we construct examples of contractions whose characteristic function F ∈ H^∞(ε →ε_∗) satisfies the conditions ‖F(z)x‖ ≥ δ‖x‖ and dim ε_∗ ⊝ F(z)ε = 1 for every z ∈ ????, x ∈ ε, but is not left invertible. In addition, we show that the condition where is the trace class of operators, which is sufficient for the left invertibility of an operatorvalued function F satisfying the estimate ‖F(z)x‖ ≥ δ‖x‖ for every z ∈ ????, x ∈ ε, with some δ > 0 (Treil, 2004), is necessary for the left invertibility of an inner function F such that dim ε_∗ ⊝ F(z)ε < ∞ for some z ∈ ????.