Non-commutative rational functions in the full Fock space

TL;DR

This paper extends classical results on rational functions in Hardy spaces to the non-commutative setting of the full Fock space, characterizing their properties, factorizations, and spectral behavior.

Contribution

It introduces a framework for understanding rational functions in the non-commutative Fock space, including factorization and spectral properties, generalizing classical Hardy space results.

Findings

Characterization of NC rational functions in the Fock space

Inner-outer factorization results for NC rational functions

Spectral analysis of NC rational multipliers

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 \emph{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.