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Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball
 
Michael A. Dritschel University of Virginia, Charlottesville
Hugo J. Woerdeman College of William & Mary, Williamsburg, VA
Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball
eBook ISBN:  978-1-4704-0200-6
Product Code:  MEMO/129/615.E
List Price: $42.00
MAA Member Price: $37.80
AMS Member Price: $25.20
Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball
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Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball
Michael A. Dritschel University of Virginia, Charlottesville
Hugo J. Woerdeman College of William & Mary, Williamsburg, VA
eBook ISBN:  978-1-4704-0200-6
Product Code:  MEMO/129/615.E
List Price: $42.00
MAA Member Price: $37.80
AMS Member Price: $25.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1291997; 62 pp
    MSC: Primary 47; 15;

    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.

    Readership

    Graduate 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 2-dilations
    • 7. Application to the inequality $|A| - \mathrm {Re}(e^{i\theta } A) \geq 0$
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1291997; 62 pp
MSC: Primary 47; 15;

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.

Readership

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 2-dilations
  • 7. Application to the inequality $|A| - \mathrm {Re}(e^{i\theta } A) \geq 0$
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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