Sub-Hardy Hilbert spaces in the non-commutative unit row-ball
Functional Analysis
2023-11-28 v2
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.
Cite
@article{arxiv.2204.05016,
title = {Sub-Hardy Hilbert spaces in the non-commutative unit row-ball},
author = {Michael T. Jury and Robert T. W. Martin},
journal= {arXiv preprint arXiv:2204.05016},
year = {2023}
}