Notes on the codimension one conjecture in the operator corona theorem

M. F. Gamal'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 33-50

Résumé

Answering a question of S. R. Treil (2004), for every $\delta$, $0\delta1$, 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 $z\in\mathbb D$.