Softcover ISBN:  9781470434557 
Product Code:  MEMO/257/1232 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
Electronic ISBN:  9781470449476 
Product Code:  MEMO/257/1232.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 257; 2019; 104 ppMSC: 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 selfadjoint contraction operators on a Hilbert space.

Table of Contents

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


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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 selfadjoint 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