On factorization of trigonometric polynomials

We give a new proof of the operator version of the Fejer-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejer-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 methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over R-m, m greater than or equal to 1, in terms of rational functions with fixed denominators.