All operator algebras have (not necessarily irreducible) boundary representations. A unital operator algebra has enough such boundary representations to generate its C *-envelope.
For R a bounded triply connected domain with boundary consisting of disjoint Jordan loops there exists an operator T on a complex Hilbert space H so that the closure of R is a spectral set for T , but T does not dilate to a normal operator with spectrum in B, the boundary of R. There is considerable overlap with the construction of an example on such a… (More)
Agler's abstract model theory is applied to C, the family of operators with unitary-dilations, where is a xed number in (0; 2]. The extremals, which are the collection of operators in C with the property that the only extensions of them which remain in the family are direct sums, are characterized in a variety of manners. They form a part of any model, and… (More)
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.… (More)
Jim Agler revolutionized the area of Pick interpolation with his realization theorem for what is now called the Agler-Schur class for the unit ball in C d. We discuss an extension of these results to algebras of functions arising from test functions and the dual notion of a family of reproducing kernels, as well as the related interpolation theorem. When… (More)
A multivariate version of Rosenblum's Fejér-Riesz theorem on outer factorization of trigonometric polynomials with operator coefficients is considered. Due to a simplification of the proof of the single variable case, new necessary and sufficient conditions for the multivariable outer factorization problem are formulated and proved.
The notion of a unitary realization is used to estimate derivatives of arbitrary order of functions in the Schur-Agler class on the polydisk and unit ball.
The Fejér-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrix-and operator-valued functions. This paper is a survey of some old and recent topics that center around Rosenblum's operator generalization of the classical Fejér-Riesz theorem.