Overview
This memoir initiates a model theory-based study of the numerical radius norm. Guided by the abstract model theory of Jim Agler, the authors propose a decomposition for operators that is particularly useful in understanding their properties with respect to the numerical radius norm. Of the topics amenable to investigation with these tools, the following are presented: A complete description of the linear extreme points of the $n\times n$ matrix (numerical radius) unit ball Several equivalent characterizations of matricial extremals in the unit ball; that is, those members which do not allow a nontrivial extension remaining in the unit ball Applications to numerical ranges of matrices, including a complete parameterization of all matrices whose numerical ranges are closed disks In addition, an explicit construction for unitary 2-dilations of unit ball members is given, Ando's characterization of the unit ball is further developed, and a study of operators satisfying $|A| - \mathrm{Re} (e^{i\theta}A)\geq 0$ for all $\theta$ is initiated.