| Abstract: | We give a new proof of the operator version of the Fejér-RieszTheorem 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 factorizationis also *-outer, is examined in greater detail. The connection to Agler smodel theory for families of operators is considered, and a set of families lyingbetween the numerical radius contractions and ordinary contractions isintroduced. The methods are also applied to the factorization of multivariateoperator-valued trigonometric polynomials, where it is shown that the factorablepolynomials are dense, and in particular, strictly positive polynomialsare factorable. These results are used to give results about factorization ofoperator valued polynomials over  , in terms of rational functionswith fixed denominators. |
