A Constructive Proof of NC Fej\'er-Riesz Theorem

TL;DR

This paper provides a constructive proof of Popescu's non-commutative Fejér-Riesz theorem, focusing on non-commuting polynomials involving multi-Toeplitz operators, advancing the theoretical understanding of non-commutative harmonic analysis.

Contribution

It offers the first constructive proof of the non-commutative Fejér-Riesz theorem for polynomials with non-commuting variables, expanding the theoretical framework.

Findings

Constructive proof of non-commutative Fejér-Riesz theorem

Application to multi-Toeplitz operators in non-commutative setting

Enhanced understanding of non-commutative polynomial positivity

Abstract

In this paper, we present a constructive proof of Popescu's non-commutative Fej\'er-Riesz theorem for non-commuting polynomials. We are considering non-commutating polynomial in left-creation and left-annihilation multi-Toeplitz operators.