Memoirs of the American Mathematical Society 1997; 62 pp; softcover Volume: 129 ISBN10: 0821806513 ISBN13: 9780821806517 List Price: US$40 Individual Members: US$24 Institutional Members: US$32 Order Code: MEMO/129/615
 This memoir initiates a model theorybased 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 2dilations 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. Readership Graduate students and research mathematicians interested in operator theory. Table of Contents  Introduction
 The Canonical Decomposition
 The Extremals \(\partial^e\)
 Extensions to the Extremals
 Linear Extreme points in \(\mathfrak C\)
 Numerical Ranges
 Unitary 2Dilations
 Application to the inequality \(A\mathrm{Re}(e^{i\theta}A)\ge 0\)
 Appendix
 References
 Index
