Sub-Hardy Hilbert spaces in the non-commutative unit row-ball

TL;DR

This paper extends classical Hardy space concepts to the non-commutative setting of the full Fock space, establishing foundational results and a Fejér-Riesz theorem for non-commutative rational functions.

Contribution

It generalizes classical Hardy space ideas to multi-variable non-commutative spaces, introducing new theoretical tools and results.

Findings

Extension of Hardy space theory to non-commutative Fock space

Development of non-commutative Fejér-Riesz theorem

Connections established between classical and non-commutative analysis

Abstract

In the classical Hardy space theory of square-summable Taylor series in the complex unit disk there is a circle of ideas connecting Szeg\"o's theorem, factorization of positive semi-definite Toeplitz operators, non-extreme points of the convex set of contractive analytic functions, de Branges--Rovnyak spaces and the Smirnov class of ratios of bounded analytic functions in the disk. We extend these ideas to the multi-variable and non-commutative setting of the full Fock space, identified as the \emph{free Hardy space} of square-summable power series in several non-commuting variables. As an application, we prove a Fej\'er-Riesz style theorem for non-commutative rational functions.