Dilations, linear matrix inequalities, the matrix cube problem, and beta distributions / J. William Helton [et al.]

Format:

Book

Author/Creator:

Helton, J. William, 1944- author.

Series:

Memoirs of the American Mathematical Society ; 1232.

Memoirs of the American Mathematical Society, 0065-9266 ; volume 257, number 1232

Language:

English

Subjects (All):

Matrices.

Matrix inequalities.

Physical Description:

1 online resource (v, 106 pages)

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, 2019.

Summary:

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.

Contents:

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.

Notes:

Description based on print version record.

Includes bibliographical references and index.

ISBN:

1-4704-4947-1