eBook ISBN:  9781470402006 
Product Code:  MEMO/129/615.E 
List Price:  $42.00 
MAA Member Price:  $37.80 
AMS Member Price:  $25.20 
eBook ISBN:  9781470402006 
Product Code:  MEMO/129/615.E 
List Price:  $42.00 
MAA Member Price:  $37.80 
AMS Member Price:  $25.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 129; 1997; 62 ppMSC: Primary 47; 15
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.
ReadershipGraduate students and research mathematicians interested in operator theory.

Table of Contents

Chapters

Introduction

1. The canonical decomposition

2. The extremals $\partial ^e$

3. Extensions to the extremals

4. Linear extreme points in $\mathfrak {C}$

5. Numerical ranges

6. Unitary 2dilations

7. Application to the inequality $A  \mathrm {Re}(e^{i\theta } A) \geq 0$


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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.
Graduate students and research mathematicians interested in operator theory.

Chapters

Introduction

1. The canonical decomposition

2. The extremals $\partial ^e$

3. Extensions to the extremals

4. Linear extreme points in $\mathfrak {C}$

5. Numerical ranges

6. Unitary 2dilations

7. Application to the inequality $A  \mathrm {Re}(e^{i\theta } A) \geq 0$