We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas
from elementary operator theory. As an outcome, an algorithm for computing the outer
polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case,
where the outer factorization is also*-outer, is examined in greater detail. The connection to
Agler's model theory for families of operators is considered, and a set of families lying
between the numerical radius contractions and ordinary contractions is introduced. The …

We prove that the sum of the entries of a positive matrix can be written as the sum of
squares. An algorithm, which allows every positive trigonometric polynomial with a
sufficiently large constant coefficient to be written as a single square of another trigonometric
polynomial, is provided. Also, this algorithm allows us to prove that every positive
trigonometric polynomial on the dual group of any abelian group, with complex (or operator)
coefficients, can be written as the difference of squares (or the sum of two squares) of …