Non-commutative rational functions in the full Fock space

Abstract:

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 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.