Regularity of loop group factorization

In the factorization of a $\operatorname{Gl}(n,\mathbb{C})$-valued loop $\varphi$ into a unitary factor and a factor holomorphic in the disk, it is shown that the two factors each have as much regularity as $\varphi$, measured in a variety of function spaces, though with exceptions. This is analogous to known results for Birkhoff factorization, but somewhat different techniques are involved.