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, r ∈ H 2 is particularly simple: The inner factor of 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 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.