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| Softcover ISBN: | 978-1-4704-3455-7 |
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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 |
| eBook ISBN: | 978-1-4704-4947-6 |
| Product Code: | MEMO/257/1232.E |
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| MAA Member Price: | $72.90 |
| AMS Member Price: | $48.60 |
| Softcover ISBN: | 978-1-4704-3455-7 |
| eBook ISBN: | 978-1-4704-4947-6 |
| Product Code: | MEMO/257/1232.B |
| List Price: | $162.00 $121.50 |
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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 self-adjoint contraction operators on a Hilbert space.
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Table of Contents
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Chapters
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1. Introduction
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2. Dilations and Free Spectrahedral Inclusions
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3. Lifting and Averaging
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4. A Simplified Form for $\vartheta $
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5. $\vartheta $ is the Optimal Bound
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6. The Optimality Condition $\alpha =\beta $ inTerms of Beta Functions
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7. Rank versus Size for the Matrix Cube
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8. Free Spectrahedral Inclusion Generalities
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9. Reformulation of the Optimization Problem
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10. Simmons’ Theorem for Half Integers
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11. Bounds on the Median and the Equipoint of the Beta Distribution
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12. Proof of Theorem
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13. Estimating $\vartheta (d)$ for Odd $d$
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14. Dilations and Inclusions of Balls
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15. Probabilistic Theorems and Interpretations Continued
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Additional Material
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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 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
