Notes on the codimension one conjecture in the operator corona theorem

Answering on the question of S.R.Treil [23], for every $\delta$, $0<\delta<1$, examples of contractions are constructed such that their characteristic functions $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ satisfy the conditions $$\|F(z)x\|\geq\delta\|x\| \ \text{ and } \ \dim\mathcal E_\ast\ominus F(z)\mathcal E =1 \ \text{ for every } \ z\in\mathbb D, \ \ x\in\mathcal E,$$ but $F$ are not left invertible. Also, it is shown that the condition $$\sup_{z\in\mathbb D}\|I-F(z)^\ast F(z)\|_{\frak S_1}<\infty,$$ where $\frak S_1$ 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, [22]), 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 $z\in\mathbb D$.