# ROOM TWO: Blaschke-Singular-Outer factorization for Hardy spaces in several non-commuting variables

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).