TY  - BOOK
AU  - Helton,J.William
AU  - Klep,Igor
AU  - McCullough,Scott
TI  - Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
T2  - Memoirs of the American Mathematical Society Series
SN  - 9781470449476
AV  - QA188 .H458 2019 
U1  - 512.9434 
PY  - 2018///
CY  - Providence
PB  - American Mathematical Society
KW  - Matrices
KW  - Matrix inequalities
KW  - Electronic books
N1  - Cover -- Title page -- Chapter 1. Introduction -- 1.1. Simultaneous dilations -- 1.2. Solution of the minimization problem (1.1) -- 1.3. Linear matrix inequalities (LMIs), spectrahedra and general dilations -- 1.4. Interpretation in terms of completely positive maps -- 1.5. Matrix cube problem -- 1.6. Matrix balls -- 1.7. Adapting the Theory to Free Nonsymmetric Variables -- 1.8. Probabilistic theorems and interpretations -- 1.9. Reader's guide -- Chapter 2. Dilations and Free Spectrahedral Inclusions -- Chapter 3. Lifting and Averaging -- Chapter 4. A Simplified Form for -- Chapter 5. þ is the Optimal Bound -- 5.1. Averages over  ( ) equal averages over  ^{ -1} -- 5.2. Dilating to commuting self-adjoint operators -- 5.3. Optimality of \ka_{*}( ) -- Chapter 6. The Optimality Condition \myal=\mybe inTerms of Beta Functions -- Chapter 7. Rank versus Size for the Matrix Cube -- 7.1. Proof of Theorem 1.6 -- Chapter 8. Free Spectrahedral Inclusion Generalities -- 8.1. A general bound on the inclusion scale -- 8.2. The inclusion scale equals the commutability index -- Chapter 9. Reformulation of the Optimization Problem -- Chapter 10. Simmons' Theorem for Half Integers -- 10.1. The upper boundary case -- 10.2. The lower boundary cases for   even -- 10.3. The lower boundary cases for   odd -- Chapter 11. Bounds on the Median and the Equipoint of the Beta Distribution -- 11.1. Lower bound for the equipoint \eiha -- 11.2. New bounds on the median of the beta distribution -- Chapter 12. Proof of Theorem 1.2 -- 12.1. An auxiliary function -- Chapter 13. Estimating þ( ) for Odd -- 13.1. Proof of Theorem 13.1 -- 13.2. Explicit bounds on þ( ) -- Chapter 14. Dilations and Inclusions of Balls -- 14.1. The general dilation result -- 14.2. Four types of balls -- 14.3. Inclusions and dilations -- Chapter 15. Probabilistic Theorems and Interpretations Continued; 15.1. The nature of equipoints -- Bibliography -- Index -- Back Cover
N2  - 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
UR  - https://ebookcentral.proquest.com/lib/orpp/detail.action?docID=5725358
ER  -