Notes on the codimension one conjecture in the operator corona theorem

Answering a question of S. R. Treil (2004), for every $\delta$, $0<\delta<1$, we constract examples of contractions whose characteristic function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ satisfies the conditions $\|F(z)x\|\geq\delta\|x\|$ and $\dim\mathcal E_\ast\ominus F(z)\mathcal E=1$ for every $z\in\mathbb D$, $x\in\mathcal E$, but is not left invertible. Also, we show that the condition $\sup_{z\in\mathbb D}\|I-F(z)^\ast F(z)\|_{\mathfrak S_1}<\infty$, where $\mathfrak S_1$ is the trace class of operators, which is sufficient for the left invertibility of the operator-valued function $F$ satisfying the estimate $\|F(z)x\|\geq\delta\|x\|$ for every $z\in\mathbb D$, $x\in\mathcal E$, with some $\delta>0$ (S. R. Treil, 2004), is necessary for the left invertibility of an inner function $F$ such that $\dim\mathcal E_\ast\ominus F(z)\mathcal E<\infty$ for some $z\in\mathbb D$.