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By classical results of Herglotz and F. Riesz, any bounded analytic functionin the complex unit disk has a unique inner-outer factorization. Here, abounded analytic function is called \emph{inner} or \emph{outer} ifmultiplication by this function defines an isometry or has dense range,respectively, as a linear operator on the Hardy Space, $H^2$, of analyticfunctions in the complex unit disk with square-summable Taylor series. Thisfactorization can be further refined; any inner function $\theta$ decomposesuniquely as the product of a \emph{Blaschke inner} function and a\emph{singular inner} function, where the Blaschke inner contains all thevanishing information of $\theta$, and the singular inner factor has no zeroesin the unit disk.  We prove an exact analog of this factorization in the context of the fullFock space, identified as the \emph{Non-commutative Hardy Space} of analyticfunctions defined in a certain multi-variable non-commutative open unit disk.

Blaschke-Singular-Outer factorization of free non-commutative functions