  A rational function belongs to the Hardy space, $H^2$, of square-summable
power series if and only if it is bounded in the complex unit disk. Any such
rational function is necessarily analytic in a disk of radius greater than one.
The inner-outer factorization of a rational function, $\mathfrak{r} \in H^2$ is
particularly simple: The inner factor of $\mathfrak{r}$ is a (finite) Blaschke
product and (hence) both the inner and outer factors are again rational.
  We extend these and other basic facts on rational functions in $H^2$ to the
full Fock space over $\mathbb{C}^d$, identified as the \emph{non-commutative
(NC) Hardy space} of square-summable power series in several NC variables. In
particular, we characterize when an NC rational function belongs to the Fock
space, we prove analogues of classical results for inner-outer factorizations
of NC rational functions and NC polynomials, and we obtain spectral results for
NC rational multipliers.
