Half-space conjecture for ranges of products of Toeplitz operators

Let $E_1$ and the relevant Hardy-space-valued symbols be as in Theorem 1, with $T_{\Phi}$ and $T_{\Theta}$ the associated Toeplitz operators. Define

$$M:=R\bigl(T_{\Phi}(I_{E_1}-T_{\Theta}T_{\Theta}^{\ast})\bigr).$$

Here a half-space is a closed subspace whose dimension and codimension are both infinite. Half-space conjecture. The subspace $M$ is a half-space if and only if $\Theta$ is not a finite Blaschke–Potapov product. The preceding proposition establishes this characterization in the cases covered there; the conjecture asserts that it holds in the full generality of Theorem 1.