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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 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 ≥ 1, in terms of rational functions with fixed denominators. Mathematics Subject Classification (2000). 47A68 (Primary), 47A65, 42A05, 30C10, 15A23 (Secondary).

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ABSTRACT. A seminal result of Agler characterizes the so-called Schur-Agler class of functions on the polydisk in terms of a unitary colligation transfer function representation. We generalize this to the unit ball of the algebra of multipliers for a family of test functions over a broad class of semigroupoids. There is then an associated interpolation theorem. Besides leading to solutions of the familiar Nevanlinna-Pick and Carathéodory-Fejér interpolation problems and their multivariable commutative and noncommutative generalizations, this approach also covers more exotic examples.

For R R a bounded triply connected domain with boundary consisting of disjoint analytic curves there exists an operator T T on a complex Hilbert space H \mathcal H so that the closure of R R is a spectral set for T T , but T T does not dilate to a normal operator with spectrum in B B , the boundary of R R . There is considerable overlap with the construction of an example on such a domain recently obtained by Agler, Harland and Raphael using numerical computations and work of Agler and Harland.