A noncommutative version of the Fej\'er-Riesz theorem

TL;DR

This paper extends the Fejér-Riesz theorem to a noncommutative setting, demonstrating that any nonnegative operator in a specific algebra can be factored as a product of an element and its adjoint.

Contribution

It introduces a noncommutative analogue of the Fejér-Riesz theorem within the algebra generated by the unilateral shift operator.

Findings

Any nonnegative operator in the algebra can be expressed as Y*Y for some Y in the algebra.

The result generalizes classical factorization results to a noncommutative framework.

Abstract

Let $\cX$ be the unital *-algebra generated by the unilateral shift operator. It is shown that for any nonnegative operator $X\in \cX$ there is an element $Y\in \cX$ such that $X=Y^*Y$.