Non-commutative rational functions in the full Fock space

A rational function belongs to the Hardy space, H2H^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 r∈H2\mathfrak {r} \in H^2 is particularly simple: The inner factor of r\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 H2H^2 to the full Fock space over Cd\mathbb {C} ^d, identified as the 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.