Non-commutative rational functions in the full Fock space

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