Notes on the codimension one conjecture in the operator corona theorem

TL;DR

This paper constructs examples of contractions with specific characteristic functions that are not left invertible, addressing a question related to the operator corona theorem and exploring conditions for invertibility.

Contribution

It provides explicit examples of contractions with characteristic functions satisfying certain bounds but lacking left invertibility, and analyzes necessary conditions for invertibility in the operator corona context.

Findings

Constructed contractions with non-invertible characteristic functions.

Showed the trace class condition is necessary for invertibility of inner functions.

Connected invertibility conditions to the operator corona theorem.

Abstract

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 $z\in\mathbb D$.