Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
Published on 30. March 2019
Book
Paperback/Softback
104 pages
978-1-4704-3455-7 (ISBN)
Description
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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 254 mm
Width: 178 mm
Weight
185 gr
ISBN-13
978-1-4704-3455-7 (9781470434557)
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Schweitzer Classification
Persons
J. William Helton, University of California, San Diego, California.
Igor Klep, The University of Auckland, New Zealand.
Scott McCullough, University of Florida, Gainesville, Florida.
Markus Schweighofer, Universitat Konstanz, Germany.
Igor Klep, The University of Auckland, New Zealand.
Scott McCullough, University of Florida, Gainesville, Florida.
Markus Schweighofer, Universitat Konstanz, Germany.
Content
Introduction
Dilations and Free Spectrahedral Inclusions
Lifting and Averaging
A Simplified Form for $\vartheta $
$\vartheta$ is the Optimal Bound
The Optimality Condition $\alpha =\beta $ in Terms of Beta Functions
Rank versus Size for the Matrix Cube
Free Spectrahedral Inclusion Generalities
Reformulation of the Optimization Problem
Simmons' Theorem for Half Integers
Bounds on the Median and the Equipoint of the Beta Distribution
Proof of Theorem 2.1
Estimating $\vartheta (d)$ for Odd $d$.
Dilations and Inclusions of Balls
Probabilistic Theorems and Interpretations continued
Bibliography
Index.
Dilations and Free Spectrahedral Inclusions
Lifting and Averaging
A Simplified Form for $\vartheta $
$\vartheta$ is the Optimal Bound
The Optimality Condition $\alpha =\beta $ in Terms of Beta Functions
Rank versus Size for the Matrix Cube
Free Spectrahedral Inclusion Generalities
Reformulation of the Optimization Problem
Simmons' Theorem for Half Integers
Bounds on the Median and the Equipoint of the Beta Distribution
Proof of Theorem 2.1
Estimating $\vartheta (d)$ for Odd $d$.
Dilations and Inclusions of Balls
Probabilistic Theorems and Interpretations continued
Bibliography
Index.