The classical Hardy space, $H^2$, is the Hilbert space of analytic functions in the complex unit disk with square-summable Taylor series coefficients at the origin. Any $h \in H^2$ has a unique inner-outer factorization $h = \theta \cdot f$, where $\theta$ is inner, i.e. multiplication by $\theta$ defines an isometry on $H^2$, and $f$ is outer, i.e. $f$ is cyclic for the shift, the isometry of multiplication by $z$. This factorization can be further refined: Any inner $\theta$ factors as $\theta = b \cdot s$ where $b$ is Blaschke, and $s$ is a singular inner. Here, the Blaschke inner factor contains all information about where (and to what degree) $\theta$ vanishes, and the singular inner factor is non-vanishing in the disk. We prove an exact analogue of this factorization in the setting of the full Fock space, identified as the Non-commutative (NC) Hardy Space of square-summable power series in several NC variables (thus refining the NC inner-outer factorization of Popescu/ Davidson-Pitts).

This is joint work with Prof. M.T. Jury (University of Florida) and Prof. E. Shamovich (Ben-Gurion University of the Negev).