VMO loop factorization equivalence theorem

Let $g\in VMO(S^1,SU(2))$. Let $A(g)$ and $A_1(g)$ be the operators associated with $g$, and let a triangular factorization mean the factorization specified in (II.3). A root subgroup factorization has the form

$$g=k_1(\eta)^*\left(\begin{matrix}e^{\chi}&0\\0&e^{-\chi}\end{matrix}\right)k_2(\zeta),$$

where $k_1$ and $k_2$ are as in Theorem 5, $\chi\in VMO(S^1;i\mathbb R)$, and $\exp(\pm\chi_+)\in L^2(S^1)$. VMO loop factorization equivalence theorem. The following are equivalent: (a) $A(g)$ and $A_1(g)$ are invertible; (b) $g$ has a triangular factorization; and (c) $g$ has the displayed root subgroup factorization. This is presented as an equivalence theorem, so the assertion is solved rather than an open conjecture.