Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions

J. William Helton, Igor Klep, Scott McCullough, Markus Schweighofer (Autoren)

Buch | Softcover

104 Seiten

2019
American Mathematical Society (Verlag)
978-1-4704-3455-7 (ISBN)

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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.

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.

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.

Erscheinungsdatum	04.02.2019
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	185 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
ISBN-10	1-4704-3455-5 / 1470434555
ISBN-13	978-1-4704-3455-7 / 9781470434557
Zustand	Neuware