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Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions

J. William Helton University of California, San Diego, California
Igor Klep The University of Auckland, Auckland, New Zealand
Scott McCullough University of Florida, Gainesville, Florida
Markus Schweighofer Universität Konstanz, Konstanz, Germany
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Softcover ISBN: 978-1-4704-3455-7
Product Code: MEMO/257/1232
List Price: $81.00 MAA Member Price:$72.90
AMS Member Price: $48.60 Electronic ISBN: 978-1-4704-4947-6 Product Code: MEMO/257/1232.E List Price:$81.00
MAA Member Price: $72.90 AMS Member Price:$48.60
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List Price: $121.50 MAA Member Price:$109.35
AMS Member Price: $72.90 Click above image for expanded view Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions J. William Helton University of California, San Diego, California Igor Klep The University of Auckland, Auckland, New Zealand Scott McCullough University of Florida, Gainesville, Florida Markus Schweighofer Universität Konstanz, Konstanz, Germany Available Formats:  Softcover ISBN: 978-1-4704-3455-7 Product Code: MEMO/257/1232  List Price:$81.00 MAA Member Price: $72.90 AMS Member Price:$48.60
 Electronic ISBN: 978-1-4704-4947-6 Product Code: MEMO/257/1232.E
 List Price: $81.00 MAA Member Price:$72.90 AMS Member Price: $48.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$121.50 MAA Member Price: $109.35 AMS Member Price:$72.90
• Book Details

Memoirs of the American Mathematical Society
Volume: 2572019; 104 pp
MSC: Primary 47; 46; 13; Secondary 60; 33; 90;

An operator $C$ on a Hilbert space $\mathcal H$ dilates to an operator $T$ on a Hilbert space $\mathcal K$ if there is an isometry $V:\mathcal H\to \mathcal K$ such that $C= V^* TV$. A main result of this paper is, for a positive integer $d$, the simultaneous dilation, up to a sharp factor $\vartheta (d)$, expressed as a ratio of $\Gamma$ functions for $d$ even, of all $d\times d$ symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.

• Chapters
• 1. Introduction
• 2. Dilations and Free Spectrahedral Inclusions
• 3. Lifting and Averaging
• 4. A Simplified Form for $\vartheta$
• 5. $\vartheta$ is the Optimal Bound
• 6. The Optimality Condition $\alpha =\beta$ inTerms of Beta Functions
• 7. Rank versus Size for the Matrix Cube
• 8. Free Spectrahedral Inclusion Generalities
• 9. Reformulation of the Optimization Problem
• 10. Simmons’ Theorem for Half Integers
• 11. Bounds on the Median and the Equipoint of the Beta Distribution
• 12. Proof of Theorem
• 13. Estimating $\vartheta (d)$ for Odd $d$
• 14. Dilations and Inclusions of Balls
• 15. Probabilistic Theorems and Interpretations Continued

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2572019; 104 pp
MSC: Primary 47; 46; 13; Secondary 60; 33; 90;

An operator $C$ on a Hilbert space $\mathcal H$ dilates to an operator $T$ on a Hilbert space $\mathcal K$ if there is an isometry $V:\mathcal H\to \mathcal K$ such that $C= V^* TV$. A main result of this paper is, for a positive integer $d$, the simultaneous dilation, up to a sharp factor $\vartheta (d)$, expressed as a ratio of $\Gamma$ functions for $d$ even, of all $d\times d$ symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.

• Chapters
• 1. Introduction
• 2. Dilations and Free Spectrahedral Inclusions
• 3. Lifting and Averaging
• 4. A Simplified Form for $\vartheta$
• 5. $\vartheta$ is the Optimal Bound
• 6. The Optimality Condition $\alpha =\beta$ inTerms of Beta Functions
• 7. Rank versus Size for the Matrix Cube
• 8. Free Spectrahedral Inclusion Generalities
• 9. Reformulation of the Optimization Problem
• 10. Simmons’ Theorem for Half Integers
• 11. Bounds on the Median and the Equipoint of the Beta Distribution
• 12. Proof of Theorem
• 13. Estimating $\vartheta (d)$ for Odd $d$
• 14. Dilations and Inclusions of Balls
• 15. Probabilistic Theorems and Interpretations Continued
Review Copy – for reviewers who would like to review an AMS book
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