The theory of rational functions in several non-commuting (NC) variables has experienced rapid growth in recent years; deep connections have been developed with established branches of pure and applied mathematics including Free Algebraic Geometry, Non-commutative Algebra, Free Probability and Control Theory. In this talk we extend classical results on rational multipliers of the Hardy space of square--summable power series in the complex unit disk to the NC setting of the Fock space, or free Hardy space, of square--summable power series in several NC variables.

Our results include: an NC rational Fej\'er--Riesz Theorem, proof that a contractive NC rational multiplier of Fock space is either inner, i.e. isometric or not an extreme point, a characterization of the NC Clark measures of both inner and contractive NC rational multipliers, as well as a constructive description of their minimal realizations and an Aronszajn--Donoghue theorem on the mutual singularity of the singular parts of the NC Clark measures of a contractive NC rational multiplier. We will show that each of these results is a natural analogue of classical fact.